Georgian Mathematical Journal Volume 9 (2002), Number 2, 227–270

AMD-NUMBERS, COMPACTNESS, STRICT SINGULARITY AND THE ESSENTIAL SPECTRUM OF OPERATORS

A. CASTEJÓN, E. CORBACHO, AND V. TARIELADZE

Abstract. For an operator T acting from an infinite-dimensional Hilbert space H to a normed space Y we define the upper AMD-number \overline{\overline{δ}}(T) and the lower AMD-number \underline{\underline{δ}}(T) as the upper and the lower limit of the net (δ(T |_{E}))_{E∈FD(H)} , with respect to the family FD(H) of all finite-dimensional subspaces of H. When these numbers are equal, the operator is called AMDregular.

It is shown that if an operator T is compact, then \overline{\overline{δ}}(T) = 0 and, conversely, this property implies the compactness of T provided Y is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator T has the property \overline{\overline{δ}}(T) = 0 if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact.

For an operator T, acting between Hilbert spaces, it is shown that \overline{\overline{δ}}(T) and \underline{\underline{δ}}(T) are respectively the maximal and the minimal elements of the essential spectrum of |T | := (T^{∗}T)^{\frac{1}{2}} , and that T is AMD-regular if and only if the essential spectrum of |T | consists of a single point.

2000 Mathematics Subject Classification: Primary: 46C05; Secondary: 47A58, 47A30. Key words and phrases: AMD-numbers, compact operator, strictly singular operator, superstrictly singular operator, Calkin algebra, essential spectrum.

1. Introduction

For a given linear operator T : E → Y, where E is an Euclidean space ( i.e., E is a non-zero finite-dimensional Hilbert space) and Y is a normed space, its mean dilatation number (briefly MD-number), δ(T) is defined by the equality

δ(T) = (∫_{S_{E}} ∥T x∥^{2}ds(x))^{\frac{1}{2}} ,  (1.1)

in which s denotes the uniform distribution on the unit sphere S_{E} of E.

The quantity δ(T), without giving it any special name, has already been used earlier in the local theory of normed spaces (see, e.g., [61, p. 81], compare also [45, p. 110]).

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The MD-number δ(T) is related to the l-norm or γ-summing norm of T through the relation (see [61, pp. 81–82])

δ(T) = \frac{1}{\sqrt{dim E}} l(T).  (1.2)

Recall that

l(T) = ( ∫_{E} ∥T x∥^{2}dγ(x))^{\frac{1}{2}} ,  (1.3)

where γ denotes the standard Gaussian distribution on E.

The l-norm was introduced in [38] and, independently, in [19] and proved to be a powerful tool of modern Functional Analysis and Operator Theory (see, e.g., [18], [10], [54], [55], [61]). It is clear that thanks to (1.2) these two functionals lead to equivalent approaches. However sometimes it is preferable to work with δ (cf., e.g., [45], [46]).

When Y is an inner product space, we have

l(T) = ∥T ∥_{HS}, δ(T) = \frac{1}{\sqrt{dim E}} ∥T ∥_{HS},  (1.4)

where ∥T ∥_{HS} denotes the Hilbert–Schmidt norm of T.

The problem of extension of the functional l to the operators acting from infinite-dimensional domain has been treated by many authors. Recall that a continuous linear operator T : H → Y, where H is an infinite-dimensional Hilbert space and Y is a normed space, is called γ-summing or Gauss-summing if

l̃(T) := sup_{M∈FD(H)} l(T |_{M}) < ∞,  (1.5)

where FD(H) stands for the family of all finite-dimensional vector subspaces M ⊂ H, dim(M) ≥ 1. This notion was introduced in [38] (see also [61, p. 82] and [55, p. 38]). It is not hard to see that when Y is a Hilbert space, then an operator T : H → Y is γ-summing if and only if it is a Hilbert–Schmidt operator and l̃(T) = ∥T ∥_{HS}. In general, the study of γ-summing operators is closely related to the problem of description of Gaussian measures in Banach spaces. In this connection (see [13], [39]), and also directly (see [38], [10], [54], [55], [61]), they were intensively studied and their relations with summing, nuclear, etc., operators were clarified.

The aim of this paper is to study appropriate asymptotic versions of MDnumbers for an operator T from an infinite-dimensional Hilbert space H to any normed space Y. Note first that always

sup_{M∈FD(H)} δ(T |_{M}) = ∥T ∥.

Therefore in this case the above described way of extending l-norm gives nothing new. Using the fact that the set FD(H) is upward directed by set-theoretic inclusion and viewing (δ(T |_{M}))_{M∈FD(H)} as a net with this index set, we assign to T two quantities:

\overline{\overline{δ}}(T) := lim sup_{M∈FD(H)} δ(T |_{M})  (1.6)

and

\underline{\underline{δ}}(T) := lim inf_{M∈FD(H)} δ(T |_{M}).  (1.7)

We call these quantities, resp., upper and lower asymptotic mean dilatation numbers (briefly upper and lower AMD-numbers) of T. In the case, where these numbers are the same, we put \overline{δ}(T) := \overline{\overline{δ}}(T) = \underline{\underline{δ}}(T) and call the operator asymptotically mean dilatation regular, briefly AMD-regular.

The notions of AMD-numbers and AMD-regularity were already considered in [12] for Hilbert space-valued operators, where it was observed, in particular, that not every operator is AMD-regular.

In this paper we study the AMD-numbers of Banach space valued operators and apply them for the investigation of compactness-like properties and the essential spectrum of operators.

The paper is organized as follows.

In Section 2 auxiliary material is collected about approximation numbers, compact operators, diagonalizable operators, strictly singular operators, Bernstein numbers and superstrictly singular operators. A proof of the following assertion, which seems not to have appeared earlier in the literature is presented: if X is a Banach space with (infra) type 2 and with an unconditional basis and Y is either an abstract L-space or a Banach space with the Orlicz property and with an unconditional basis, then any strictly singular operator T : X → Y is compact (Theorem 2.18).

Section 3 is dedicated to the MD-numbers. Among other, rather technical, inequalities, it is shown, in particular, that the MD-number δ(T) and the median of T with respect to the uniform distribution are equivalent quantities (see Proposition 3.7 and the Remark following it). The most delicate result is Proposition 3.9 which in fact is related to the Isoperimetric Inequality.

In Section 4 the general properties of AMD-numbers are analyzed. In the case of diagonalizable operators, i.e., operators having the form T = ∑ λ_{n}e_{n} ⊗ y_{n}, where (λ_{n}) is a bounded sequence of scalars and (e_{n}) and (y_{n}) are orthonormal bases in H and Y, respectively, the following concrete expressions of these numbers are obtained:

\overline{\overline{δ}}(T) = lim sup_{n} |λ_{n}|, \underline{\underline{δ}}(T) = lim inf_{n} |λ_{n}|.

From this result it follows that diagonalizable T with diagonal (λ_{n}) is AMDregular if and only if the sequence (|λ_{n}|) is convergent and when this is the case, we have

\overline{δ}(T) = lim_{n} |λ_{n}|.

This assertion implies, in particular, that not every operator is AMD-regular (cf. [12], where it is shown directly that for any partial isometry T with infinitedimensional initial and final spaces one has \overline{\overline{δ}}(T) = 1, while \underline{\underline{δ}}(T) = 0).

In the final part of Section 4 AMD-numbers are applied for characterization of compactness, strict singularity and superstrict singularity of operators. It is shown, e.g., that if T is a compact operator, then \overline{\overline{δ}}(T) = 0 and that the converse statement is also true provided Y has cotype 2, but may not be true without this requirement. More precisely, it turns out that for a given operator T, in general, \overline{\overline{δ}}(T) ≤ d(T, K(H, Y)) and if Y is of cotype 2, then \overline{\overline{δ}}(T) ≥ c^{−1}d(T, K(H, Y)), where c is the cotype 2 constant of Y and d(T, K(H, Y)) stands for the distance from T to the space of all compact operators K(H, Y). The core of the section is Theorem 4.11 which establishes, in particular, that in case of an arbitrary Y for a given T the fulfilness of the condition \overline{\overline{δ}}(T) = 0 is equivalent to the superstrict singularity of T. This result shows that the upper AMD-number \overline{\overline{δ}}(T) can also be viewed as a “distance” from the operator T to the set of superstrictly singular operators. The section ends with the following “mean dilatation free” consequence of the previous results: any superstrictly singular operator from a Hilbert space to an arbitrary cotype 2 Banach space is compact (Theorem 4.12).

Section 5 deals with the operators between Hilbert spaces. The general case of non-necessarily diagonalizable operators is treated. We prove that for a given operator T, \overline{\overline{δ}}(T) is the maximal and \underline{\underline{δ}}(T) is the minimal element of the essential spectrum of |T | := (T^{∗}T)^{\frac{1}{2}} and that T is asymptotically mean dilatation regular if and only if the essential spectrum of |T | consists of a single point (Theorem 5.6).

We derive this result from the corresponding statement concerning diagonalizable operators. For this several assertions, about the continuity of a spectrum and spectral radius in a Banach algebra, essential spectrum, etc. are used. Surely, most of these facts are well-known for specialists, we give their precise formulations and outline the proofs only for the reader’s convenience.

2. Auxiliary Results on Compact and Diagonalizable Operators

2.1. Notation. Hereinafter 𝕂 denotes either the field ℝ of real numbers or the field ℂ of complex numbers. The considered normed or inner-product spaces are supposed to be defined over 𝕂. The norm of a normed space, resp., the scalar product of an inner-product space is denoted by ∥ · ∥, resp., (·|·). Also, ∥T ∥ stands for the ordinary norm of a continuous linear operator T acting between normed spaces.

For a normed space X, • X^{∗} stands for the topological dual space.

• We put:

B_{X} := {x ∈ X : ∥x∥ ≤ 1}, S_{X} := {x ∈ X : ∥x∥ = 1}.

• For x ∈ X and a non-empty A ⊂ X

d(x, A) := inf{∥x − a∥ : a ∈ A}.

• FD(X) stands for the family of all finite-dimensional non-zero vector

subspaces of X. For normed spaces X and Y,

• L(X, Y) is the normed space of all continuous linear operators T : X → Y and L(X) := L(X, X).

• K(X, Y) stands for the set of all compact linear operators T : X → Y and K(X) := K(X, X).

• d(T, K(X, Y)) := inf{∥T − A∥ : A ∈ K(X, Y), T ∈ L(X, Y)}.

• For an operator T ∈ L(X, Y) we put

m(T) := inf{∥T x∥ : x ∈ S_{X}}

and call the quantity the lower bound of T.

• rank T := dim T (X), T ∈ L(X, Y). By H we always denote an infinite-dimensional (not necessarily separable)

Hilbert space, which is canonically identified with H^{∗}. In particular, when Y is also a Hilbert space, for a given operator T ∈ L(H, Y) its adjoint operator T^{∗} is supposed to act from Y to H.

For Hilbert spaces H, Y and an operator T ∈ L(H, Y) the unique self-adjoint positive (=non-negative) square root of the operator T^{∗}T ∈ L(H) is denoted by |T |.

Let J be an infinite abstract index set and (β_{j})_{j∈J} be a family of elements of a topological space. Suppose in J a directed partial order ≺ is given. Then the family (β_{j})_{j∈(J,≺)} is a net with respect to (J, ≺) and, for it, the notion of convergence and the meaning of lim_{j∈(J,≺)} β_{j} are clear. If (β_{j})_{j∈J} is a family of real

numbers, then the meaning of notations lim sup_{j∈(J,≺)} β_{j} and lim inf_{j∈(J,≺)} β_{j} is also clear:

lim sup_{j∈(J,≺)} β_{j} := inf_{j∈J} sup_{i∈J, i≻j} β_{j}, lim inf_{j∈(J,≺)} β_{j} := sup_{j∈J} inf_{i∈J, i≻j} β_{j}.

Below we shall use the notation of an ‘upper limit’, a ‘lower limit’ and a ‘limit’ of a given family (β_{j})_{j∈J} of real numbers, which do not assume that some directed partial order is given in J, namely, we put :

u-lim sup_{j∈J} β_{j} := inf_{Δ⊂J,card Δ<∞} sup_{j∉Δ} β_{j}, u-lim inf_{j∈J} β_{j} := sup_{Δ⊂J,card Δ<∞} inf_{j∉Δ} β_{j}.

The letter ‘u’ means ‘unordered’^{1}. When we have that u-lim sup_{j} β_{j} = ulim inf_{j} β_{j} = β, then we say that the unordered limit of (β_{j})_{j∈J} exists and put

u-lim_{j∈J} β_{j} = β.

1 Let 𝔊 be the filter generated by complements of finite subsets of J. Following [7, Ch. IV, §5], instead of u-lim sup_{j∈J} β_{j} we could write lim sup_{𝔊} β_{j} and call this quantity “the upper limit with respect to the filter 𝔊”.

The following statement is easy to check.

Lemma 2.1. Let (β_{j})_{j∈J} be any bounded infinite family of real numbers.

(a) We have the equalities:

u-lim sup_{j∈J} β_{j} = sup_{(j_{n})} lim_{n} β_{j_{n}} and u-lim inf_{j∈J} β_{j} = inf_{(j_{n})} lim_{n} β_{j_{n}},

where sup, (resp. inf), is taken over all infinite sequences (j_{n}) of distinct elements of J such that the sequence (β_{j_{n}})_{n∈ℕ} is convergent in ℝ.

(b) If J = ℕ, then

u-lim sup_{j∈ℕ}β_{j} = lim sup_{j→∞} β_{j} and u-lim sup_{j∈ℕ}β_{j} = lim inf_{j→∞} β_{j}.

In particular, lim_{j→∞} β_{j} exists if and only if u-lim_{j∈ℕ}β_{j} exists and in such a case

lim_{j→∞} β_{j} = u-lim_{j∈ℕ}β_{j}.

For a given bounded infinite family of scalars λ_{·} := (λ_{j})_{j∈J} and for any fixed natural number n we put

a_{n}(λ_{·}):= inf_{Δ⊂J, card Δ<n} sup{|λ_{j}|j ∈J \Δ}, ã_{n}(λ_{·}):= sup_{Δ⊂J, card Δ=n} min{|λ_{j}| : j ∈ Δ}.

Lemma 2.2. Let λ_{·} := (λ_{j})_{j∈J} be a bounded infinite family of scalars. Then:

(a) We have a_{n}(λ_{·}) = ã_{n}(λ_{·}), n = 1, 2, …

(a′) If J_{+} := {j ∈ J : |λ_{j}| > 0} ≠ ∅ and λ_{·}^{+} := (λ_{j})_{j∈J_{+}}, then a_{n}(λ_{·}) = a_{n}(λ_{·}^{+}), n = 1, 2, …

(b) (a_{n}(λ_{·})) is a decreasing sequence and u-lim sup_{j∈J} |λ_{j}| = lim_{n} a_{n}(λ_{·}).

(c) If J = ℕ and (|λ_{j}|)_{j∈ℕ} is a decreasing sequence, then a_{n}(λ_{·}) = |λ_{n}|, n = 1, 2, …

Proof. (a) Fix n and let ã_{n}(λ_{·}) < r < ∞. Then for each Δ ⊂ J with card(Δ) = n there is j ∈ Δ such that |λ_{j}| < r. Denote J_{r} = {j ∈ J : |λ_{j}| ≥ r}. Then card J_{r} < n and sup{|λ_{j}| : j ∈ J \ J_{r}} < r. Hence a_{n}(λ_{·}) < r. Since r is arbitrary, we get a_{n}(λ_{·}) ≤ ã_{n}(λ_{·}). The proof of a_{n}(λ_{·}) ≥ ã_{n}(λ_{·}) is similar.

(a′) is evident.

We omit a straightforward verification of other statements.^{2}

Denote by 𝔉(J) the family of all non-empty finite subsets of a set J, then (𝔉(J), ⊂) is a directed set. A family (x_{j})_{j∈J} of elements of a Banach space X is called summable with sum x ∈ X, in symbols ∑_{j∈J} x_{j} = x, if the net (∑_{j∈Δ} x_{j})_{Δ∈𝔉(J)} is convergent to x in the topology of X.

It is well-known that a sequence (x_{j})_{j∈ℕ} is summable if and only if the corresponding series is unconditionally convergent and in such a case ∑_{j∈ℕ} x_{j} = ∑_{j=1}^{∞} x_{j}. Note also that if (x_{j})_{j∈J} is a summable family, then the set J_{+} of those indices j for which x_{j} ≠ 0 is at most countable and ∑_{j∈J} x_{j} = ∑_{j∈J_{+}} x_{j}.

2 It is not hard to check that for a given bounded family (λ_{j})_{j∈J} the quantity a_{n}(λ_{·})

coincides with its n-th approximation number in the sense of [52, 13.7.3].

A family (y_{j})_{j∈J} of elements of a Banach space Y is called an unconditional basis of Y if for any y ∈ Y there exists a unique family (t_{j})_{j∈J} of scalars such that y = ∑_{j∈J} t_{j}y_{j}.

If (y_{j})_{j∈J} is an unconditional basis of Y, then in Y^{∗} there exists a family of coordinate functionals (y_{j}^{∗})_{j∈J} such that ∑_{j∈J} y_{j}^{∗}(y)y_{j} = y, ∀y ∈ Y.

A family (y_{j})_{j∈J} of elements of a Banach space Y is called an unconditional basic family if it is an unconditional basis for its closed linear span into Y.

Note that if Y is a separable infinite-dimensional Banach space and (y_{j})_{j∈J} is an infinite unconditional basic family in it, then J is countable.^{3}

Let Y be a Hilbert space. Recall that for an operator T ∈ L(H, Y) the value of the sum ∑_{j∈J} ∥T e_{j}∥^{2} does not depend on a particular choice of an orthonormal basis (e_{j})_{j∈J} of H and when this value is finite, the operator is called a Hilbert–Schmidt operator. It is well known that the equality

∥T ∥_{HS} = (∑_{j∈J} ∥T e_{j}∥^{2})^{\frac{1}{2}}  (2.1)

defines a norm on the vector space of all Hilbert–Schmidt operators. This norm is called the Hilbert–Schmidt norm.

2.2. Approximation numbers, Bernstein numbers and compact-like operators. Let X, Y be normed spaces. A linear operator T : X → Y is called compact if T (B_{X}) is a relatively compact subset of Y.

A linear operator T : X → Y is called completely continuous if for any weak-null sequence (x_{n}) in X, lim_{n} ∥T x_{n}∥ = 0. Any T ∈ K(X, Y) is completely continuous; the converse is true if X is reflexive (or is ‘almost reflexive’, see Proposition 2.3), but is not true in general (see the next item).

A normed space X is said to have the Schur property if for any weak-null sequence (x_{n}) in X, we have lim_{n} ∥x_{n}∥ = 0 (i.e., if the identity mapping I : X → X is completely continuous). The space X = l_{1} is a classical example of an infinite-dimensional Banach space with the Schur property.

Proposition 2.3 (Rosenthal, Lacey–Whitley). Let X be a Banach space which does not contain a subspace isomorphic to ℓ_{1} and Y be another Banach space.

(a) Any completely continuous linear operator T : X → Y is compact.

(b) If Y has the Schur property, then L(X, Y) = K(X, Y).

Proof. According to Rosenthal’s ℓ_{1}-theorem [17, p. 201] each bounded sequence in X has a weak-Cauchy subsequence, i.e., X is almost reflexive in the sense [36]. For almost reflexive X the assertion coincides with Theorem 5 and Corollary 6 in [36].

3 Every infinite-dimensional Banach space contains a basic sequence [40, Theorem 1.a.5]; however, the famous unconditional basic sequence problem “Does every infinite dimensional Banach space contain an unconditional basic sequence?” [40, Problem 1.d.5] has been solved negatively in [24].

Remark 2.4. The following converse to Proposition 2.3(a) is also true: suppose X is a Banach space such that any completely continuous linear operator T : X → c_{0} is compact, then X does not contain a subspace isomorphic to ℓ_{1} (see [4, Proposition II.6]).

For a given operator T ∈ L(X, Y) and a natural number n, the n-th approximation number a_{n}(T) of T is defined by the equality:

a_{n}(T) = inf{∥T − T_{0}∥ : T_{0} ∈ L(X, Y), rank T_{0} < n}.

It is easy to see that (a_{n}(T)) is a decreasing sequence and a_{1}(T) = ∥T ∥ (see [52] and [53] for further properties).

Remark 2.5. An operator T ∈ L(X, Y) is called approximable if lim_{n} a_{n}(T) = 0. If Y is a Banach space, then any approximable T ∈ L(X, Y) is compact. The converse is also true provided either X^{∗} or Y has the approximation property (see [40] for the definition and proofs).

Lemma 2.6 ([55, Lemma 1.8 (p.10)]). Let Y be a normed space and T ∈ L(H, Y). Then for every ε > 0 there is an orthonormal sequence (e_{n}) in H such that ∥T e_{n}∥ ≥ a_{n}(T) − ε for all n ≥ 1.

The next statement characterizes the compact operators acting from a Hilbert space.

Corollary 2.7. Let Y be a normed space and T ∈ L(H, Y). Then the following assertions are equivalent:

(i) For any infinite orthonormal sequence (e_{n}) in H,lim_{n} ∥T e_{n}∥ = 0.

(ii) lim_{n} a_{n}(T) = 0.

(iii) The operator T is the norm limit of a sequence of finite-rank operators from L(H, Y).

(iv) The operator T is compact.

(v) The operator T is completely continuous.

Proof. (i) ⇒ (ii) follows from Lemma 2.6. The implications (ii) ⇒ (iii) and (iv) ⇒ (v)⇒ (i) are evident. (iii) ⇒ (iv) is well-known, when Y is complete. In general, (iii) implies that T (B_{H}) is precompact in Y; since B_{H} is weakly compact and T is weakly continuous too, T (B_{H}) is also weakly compact in Y. These two conditions give that T (B_{H}) is compact in Y (see [8, IV, §1, Proposition 3]).

Corollary 2.8. Let T ∈ L(H, Y). Then

(a) lim_{n} a_{n}(T) = d(T, K(H, Y)).

(b) d(T, K(H, Y)) = sup_{(e_{n})∈O(H)} lim sup_{n} ∥T e_{n}∥, where O(H) denotes the set of all infinite orthonormal sequences (e_{n}) in H.

Proof. (a) follows easily from the implication (iv) ⇒ (iii) of Corollary 2.7. (b) The inequality d(T, K(H, Y)) ≤ sup_{(e_{n})∈O(H)} lim sup_{n} ∥T e_{n}∥ follows from (a) and Lemma 2.6. The inequality d(T, K(H, Y)) ≥ sup_{(e_{n})∈O(H)} lim sup_{n} ∥T e_{n}∥ can be shown as follows: fix any (e_{n}) ∈ O(H) and A ∈ K(H, Y). Clearly (e_{n}) is weak-null in H. Since A is compact, we have lim_{n} ∥Ae_{n}∥ = 0. Consequently, ∥T − A∥ ≥ lim sup_{n}(∥T e_{n}∥ − ∥Ae_{n}∥) = lim sup_{n} ∥T e_{n}∥.

Remark 2.9. The same proof shows that if X is an arbitrary normed space without the Schur property, then d(T, K(X, Y)) ≥ lim sup_{n} ∥T x_{n}∥ for any normalized weak-null sequence (x_{n}) in X (cf. [58, Lemma 2(i)]).

An operator T ∈ L(X, Y) is called strictly singular (briefly, SS-) if m(T |_{M}) = 0 for any infinite-dimensional closed vector subspace M ⊂ X. Denote SS(X, Y) the set of all strictly singular operators T : X → Y and SS(X) := SS(X, X).

We have K(X, Y) ⊂ SS(X, Y). In general, this inclusion is strict (see Remark 2.11).

Fix T ∈ L(X, Y) and a natural number n. The n-th Bernstein number b_{n}(T) of T is defined by the equality (see [48]):

b_{n}(T) = sup{m(T |_{M}) : M ∈ FD(X), dim M = n}.  (2.2)

Obviously, ∥T ∥ = b_{1}(T) ≥ b_{2}(T) ≥ . . .. If T : X → Y is a compact operator, then lim_{n} b_{n}(T) = 0, the converse is not true in general (see the next item and Proposition 2.10(c)).

An operator T ∈ L(X, Y) is called superstrictly singular (briefly, SSS-) if lim_{n} b_{n}(T) = 0. Denote SSS(X, Y) the set of all superstrictly singular operators T : X → Y and SSS(X) := SSS(X, X).

We have K(X, Y) ⊂ SS(X, Y) ⊂ SSS(X, Y); in general, the second inclusion also is strict (see Proposition 2.10(e)). It is known that the strictly singular operators and the superstrictly singular form an operator ideal in the sense of Pietsch (see [52, (1.9.4)] and [56]).

In the next proposition are collected mainly the known statements.

Proposition 2.10. Let 1 ≤ r, p < ∞.

(a) If r < p, then L(l_{p}, l_{r}) = K(l_{p}, l_{r}) (Pitt’s theorem; [40, p. 76]).

(b) If X and Y are totally incomparable Banach spaces, then any continuous linear operator from X to Y is strictly singular [40, p. 75].

In particular,

If 1 ≤ r, p < ∞ and r ≠ p, then SS(l_{r}, l_{p}) = L(l_{r}, l_{p}) (see [23]).

(c) The natural embedding T : l_{1} → l_{2} is a SSS-operator that is not compact [48].

(d) If 1 ≤ r < p < ∞ and T is the natural embedding of l_{r} into l_{p}, then T ∈ SSS(l_{r}, l_{p}) \ K(l_{r}, l_{p}) (see [44, Lemma 1]).^{4}

(e) If 1 < r < p < ∞, then SSS(l_{r}, l_{p}) ≠ SS(l_{r}, l_{p}).^{5} Proof. (e) For a fixed number s, 1 < s < ∞, let us denote by X_{s} the l_{s}-sum of spaces l_{2}^{n}, n = 1, 2, . . .. According to a well-known result of [49] (see also [40, p. 73]), there exists a bijective linear homeomorphism U_{s} : X_{s} → l_{s}. Let T_{r,p} be the natural embedding of X_{r} into X_{p}. It is clear that T_{r,p} ∉ SSS(X_{r}, X_{p}), hence U_{p}T_{r,p}U_{r}^{−1} ∉ SSS(l_{r}, l_{p}), while, by (b), we have U_{p}T_{r,p}U_{r}^{−1} ∈ SS(l_{r}, l_{p}).

4 Cf. also [56], where it is shown that in the considered case one has the equalities b_{n}(T) = n^{1/p−1/r}, n = 1, 2, …, which imply (d) too.

5 The third author learnt this result, as well as its proof, from Prof. A. Pełczyński.

Remark 2.11. (1) The notion of a strictly singular operator was introduced in [30], where it was established that SS(X, Y) is a closed vector subspace of L(X, Y); in the case of Hilbert spaces it was shown the equality K(X, Y) = SS(X, Y) and a question was posed whether the same is true in the general case (see [30, p. 285]). Later in [21] it was shown that K(c_{0}) = SS(c_{0}) and K(l_{p}) = SS(l_{p}), ∀p ∈ [1, ∞[ and an example was given (due to M. I. Kadets, see [21, p. 61]) showing that this is not true in general.

(2) The SSS-operators were introduced in fact in [48], where no particular term was given to this notion. The term is taken from [28]. For this notion the term “𝔖_{0}^{∗}-operators” or “operators of class C_{0}^{∗}” was used in [42], [43], [44].^{6}

(3) Numerous papers were devoted to the study of strictly singular operators, strictly cosingular operators (introduced in [50]), disjointly strictly singular operators (introduced in [26]) and related operators (see, e.g., [3], [9], [11], [14], [15], [20], [23], [27], [36], [31], [42], [43], [44], [47], [48], [50], [51], [56], [64]). Some known facts are already included in the monographs (see [22], [40], [52]). As far as we know, the most recent work which deals with the superstrictly singular operators is [56].

Below we present a result about automatic compactness of strictly singular operators.

2.3. Orlicz property, type 2 and SS-operators. In this subsection X, Y will be Banach spaces.

Let us say that an (infinite) family (y_{j})_{j∈J} of elements of a Banach space Y is Besselian if there exists a constant b > 0 such that the inequality

b∥ ∑_{j∈J} t_{j}y_{j}∥ ≥ ( ∑_{j∈J} |t_{j}|^{2})^{\frac{1}{2}}  (2.3)

holds for any family (t_{j})_{j∈J} of scalars with only a finite number of non-zero terms.^{7}

For a Besselian family (y_{j})_{j∈J} any constant b for which (2.3) holds is called its Besselian constant.

Analogously, let us say that an (infinite) family (y_{j})_{j∈J} of elements of a Banach space Y is Hilbertian if there exists a constant b > 0 such that the inequality

6 As the referee has pointed out, the ”uniformly strictly singular operators” from [47] are different from SSS-operators.

7 Sometimes a Besselian sequence is called a 2-colacunary sequence, see [1].

∥ ∑_{j∈J} t_{j}y_{j}∥ ≤ b( ∑_{j∈J} |t_{j}|^{2})^{\frac{1}{2}}  (2.4)

holds for any family (t_{j})_{j∈J} of scalars with only a finite number of non-zero terms.

For a Hilbertian family (y_{j})_{j∈J} any constant b for which (2.4) holds is called its Hilbertian constant.

In what follows, the set {−1, 1}^{ℕ} will be supposed to be equipped with the probability measure ∏_{n∈ℕ} μ_{n}, where μ_{n}\{−1\} = μ_{n}\{1\} = \frac{1}{2}, n = 1, 2, …^{8} Recall that a Banach space Y is said to have the

• Orlicz property if for any summable sequence (y_{j})_{j∈ℕ} in Y, we have ∑_{j=1}^{∞} ∥y_{j}∥^{2} < ∞.

• Rademacher cotype 2 if for every sequence (y_{j})_{j∈ℕ} in Y for which the series ∑_{j∈ℕ} θ_{j}y_{j} is convergent in Y for almost all choices of signs (θ_{j}) ∈ {−1, 1}^{ℕ}, we have ∑_{j=1}^{∞} ∥y_{j}∥^{2} < ∞.

• infratype 2 if for any sequence (y_{j})_{j∈ℕ} in Y, with ∑_{j=1}^{∞} ∥y_{j}∥^{2} < ∞ there exists a sequence of signs (θ_{j}) ∈ {−1, 1}^{ℕ} such that the series ∑_{j∈ℕ} θ_{j}y_{j} is convergent in Y.

• Rademacher type 2 if for any sequence (y_{j})_{j∈ℕ} in Y, with ∑_{j=1}^{∞} ∥y_{j}∥^{2} < ∞, the series ∑_{j∈ℕ} θ_{j}y_{j} is convergent in Y for almost all choices of signs (θ_{j}) ∈ {−1, 1}^{ℕ}.

Remark 2.12. Let Y be an infinite-dimensional Banach space.

(1) Clearly, if Y has the Rademacher cotype 2, then Y has the Orlicz property. If Y is a rearrangement invariant (=symmetric) Banach function space on [0, 1], then, according to [59, Theorem 1], the converse is also true. However, the converse is not true in general: in [60] an example of a Banach space with an unconditional (even with a symmetric) basis is produced, which satisfies the Orlicz property and fails to be of cotype 2.

(2) Also is clear that if Y has the Rademacher type 2, then Y has infratype 2. Whether the converse is true is not known.

(3) It is standard to see that if Y has the Orlicz property, then in Y any unconditional normalized basic sequence is Besselian.

(4) Also, it is easy to see that if Y has infratype 2, then in Y any unconditional normalized basic sequence is Hilbertian.

(5) If Y has infratype 2, then Y does not contain a subspace isomorphic to ℓ_{1} (since ℓ_{1} has no infratype 2).

(6) If Y has infratype 2, then Y has no Schur property ( this follows from (5) and from the fact that any infinite-dimensional Banach space with the Schur property contains a subspace isomorphic to ℓ_{1} [17, p. 212]).

8 This is simply the normalized Haar measure of the compact multiplicative group {−1, 1}^{ℕ}.

Let us say that a Banach space Y possesses the Besselian Selection Property, shortly, the BS-property, if in Y any weak-null sequence (y_{n})_{n∈ℕ} with inf_{n∈ℕ} ∥y_{n}∥ > 0 has a Besselian basic subsequence.^{9}

We say that a Banach space Y possesses the Hilbertian Selection Property, shortly, the HS-property, if Y has no Schur property and in Y any weak-null sequence (y_{n})_{n∈ℕ} with inf_{n∈ℕ} ∥y_{n}∥ > 0 has a Hilbertian basic subsequence.

Proposition 2.13. Let Y be an infinite-dimensional Banach space.

(a) Suppose that Y has the Orlicz property and has a (not necessarily countable) unconditional basis. Then in Y any weak-null sequence (y_{n})_{n∈ℕ} with inf_{n∈ℕ} ∥y_{n}∥ > 0 has a Besselian unconditional basic subsequence.

In particular, Y possesses the BS-property.

(b) [1, Theorem 6] If Y is any (abstract) L-space, then Y possesses the BSproperty. ^{10}

(c) Suppose Y has infratype 2 and has a (not necessarily countable) unconditional basis. Then in Y any weak-null sequence (y_{n})_{n∈ℕ} with inf_{n∈ℕ} ∥y_{n}∥ > 0 has a Hilbertian unconditional basic subsequence.

In particular, Y possesses the HS-property.

(d) If Y = c_{0}, then Y possesses the HS-property.

Proof. (a,c) Fix an unconditional basis (e_{j})_{j∈J} of Y and let (y_{n})_{n∈ℕ} be a weaknull sequence in Y with inf_{n∈ℕ} ∥y_{n}∥ > 0. There is a countable J_{0} ⊂ J such that y_{n} ∈ Y_{0}, n = 1, 2, …, where Y_{0} is the closed subspace of Y generated by (e_{j})_{j∈J_{0}}.

According to Bessaga–Pelczyński’s selection principle (see [17, p. 46]) (y_{n})_{n∈ℕ} has a subsequence (y_{k_{n}}), which is equivalent to a block basic sequence of the unconditional basis (e_{j})_{j∈J_{0}} of Y_{0}. Consequently, (y_{k_{n}}) also is an unconditional basic sequence. By Remark 2.12(3) (resp.(4)) we get that (y_{k_{n}}) is a Besselian (resp. Hilbertian) unconditional basic sequence.

(d) is true by the same reasoning as (c) because the natural basis of c_{0} is Hilbertian and equivalent to any normalized block basic sequence taken with respect to it [40, Proposition 2.a.1].

Remark 2.14. (1) The statement (a) of Proposition 2.13 for a separable Banach Y space with Rademacher cotype 2 and with an unconditional basis follows at once from [1, Proposition 9].

(2) The statement (b) of Proposition 2.13 is applicable, e.g., for the cotype 2 space Y = L_{1}[0, 1], while (a) is not because this space has not an unconditional basis.

9 Formally, we get that if Y has the Schur property, then Y possesses the BS-property.

Actually, if Y has the Schur property, then it possesses a stronger property: any bounded sequence (y_{n})_{n∈ℕ} in Y, which has no convergent subsequence, admits a basic subsequence, which is equivalent to the natural basis of l_{1} (by Rosenthal’s theorem [17, p. 201]).

10 Actually, this is an easy consequence of Theorem 6 in [1], which asserts more: if Y is an L-space and (y_{n})_{n∈ℕ} is a bounded sequence in Y, then either (y_{n})_{n∈ℕ} has a convergent subsequence or (y_{n})_{n∈ℕ} has a Besselian subsequence. We are grateful to Professors A. Pełczyński and H.P. Rosenthal for paying our attention to the paper [1].

(3) It is not clear whether any Y with the Orlicz property (or with Rademacher cotype 2) possesses the BS-property (cf. [1, p. 304, Remark]).

Lemma 2.15. Let X be a Banach space possessing the HS-property, Y be a Banach space possessing the BS-property and T ∈ L(X, Y) be an operator which is not completely continuous. Then

(a) there exists a Hilbertian basic sequence (x_{n}) in X such that (T x_{n}) is a Besselian basic sequence in Y;

(b) there exists an infinite-dimensional closed vector subspace M ⊂ X such that M is isomorphic to ℓ_{2} and m(T |_{M}) > 0.

Proof. (a) Since T is not completely continuous, there exists a normalized weaknull sequence (z_{n}) in X such that inf_{n} ∥T z_{n}∥ := r > 0. Clearly, (T z_{n}) is a weak-null sequence in Y. Since Y possesses the BS-property, (T z_{n}) has a Besselian basic subsequence (T z_{k_{n}}). Let a_{n} := z_{k_{n}}, n = 1, 2, . . .. Now, since X has the HS-property, the sequence (a_{n}) has a Hilbertian basic subsequence (a_{m_{n}}) = (x_{n}). Evidently, the sequence (x_{n}) has the needed properties.

(b) Let (x_{n}) be a sequence from (a). Denote M the closed vector subspace of X generated by (x_{n}). Let also b_{1} (resp. b_{2}) be a Hilbertian (resp. a Besselian) constant of (x_{n}) (resp. of (T x_{n})). Since (T x_{n}) is Besselian, we get that (x_{n}) is Besselian too (with constant b_{2} ∥T ∥). Therefore the basis (x_{n}) of M is both Hilbertian and Besselian. Consequently M is isomorphic to ℓ_{2}.

Fix x ∈ M. Then for a sequence (t_{j}) of scalars we have x = ∑_{j=1}^{∞} t_{j}x_{j} and

∥T x∥ = ∥∑_{j=1}^{∞} t_{j}T x_{j}∥ ≥ b_{2}^{−1} (∑_{j=1}^{∞} |t_{j}|^{2})^{1/2} ≥ b_{2}^{−1}b_{1}^{−1}∥x∥.

Hence, m(T |_{M}) ≥ b_{2}^{−1}b_{1}^{−1} > 0.

Corollary 2.16. Let X be a Banach space possessing the HS-property and Y be a Banach space possessing the BS-property. Then we have:

(a) Any T ∈ SS(X, Y) is completely continuous.

(b) If either X or Y does not contain a closed vector subspace isomorphic to ℓ_{2}, then any T ∈ L(X, Y) is completely continuous.

Proof. (a,b) Suppose there exists T ∈ L(X, Y) which is not completely continuous. Then by Lemma 2.15(b) there is an infinite-dimensional closed subspace M ⊂ X such that M is isomorphic to ℓ_{2} and m(T |_{M}) > 0. Hence, T ∉ SS(X, Y) and we get (a). Since m(T |_{M}) > 0, the subspace T (M) of Y is isomorphic to M. Therefore X and Y both have a subspace isomorphic to ℓ_{2} and we get (b).

Remark 2.17. The following example shows that in Corollary 2.16(a) it cannot be asserted that any T ∈ SS(X, Y) is compact. Let X = ℓ_{1}×c_{0} and Y = ℓ_{2}. Then X possesses the HS-property (this is easy to see). Let now T = T_{1,2}P_{1}, where T_{1,2} : ℓ_{1} → ℓ_{2} is the natural embedding and P_{1} : ℓ_{1}×c_{0} → ℓ_{1} is the natural projection. Then T is even superstrictly singular (as T_{1,2} has this property), but T is not compact.

We can formulate the result which seems not to have been noticed in the literature.

Theorem 2.18. Let X be a Banach space of infratype 2 with an unconditional basis, Y be either a Banach space having the Orlicz property and an unconditional basis or any abstract L-space. Then any strictly singular operator T : X → Y is compact.

Proof. Observe that by Proposition 2.13(c) X possesses the HS-property and by Proposition 2.13(b,c) Y possesses the BS-property. Let now T ∈ SS(X, Y). Then by Corollary 2.16(a), T is completely continuous. Since X is of infratype 2, it does not contain a closed vector subspace isomorphic to ℓ_{1} (see Remark 2.12(5)). Then by Proposition 2.3(a) we get that T is a compact operator.

Remark 2.19. (1) Theorem 2.18 implies, in particular, that K(H) = SS(H).

(2) When X = ℓ_{2} and Y is a cotype 2 space with unconditional basis, Theorem 2.18 was pointed out to us by Prof. A. Plichko.

(3) In [11] a general result is obtained, from which it follows that if 1 < r ≤ p < ∞, then any regular strictly singular T : 𝕃_{p}[0, 1]) → 𝕃_{r}[0, 1] is compact; Theorem 2.18 implies that if 1 ≤ r ≤ 2 ≤ p ≤ ∞, then the same conclusion is true without assuming additionally that T is regular.

(4) Theorem 2.18 leaves the following question open: let Y be an arbitrary Banach space with Orlicz property (or with Rademacher cotype 2); is then any strictly singular operator T : l_{2} → Y compact? Below, through AMD-numbers, we obtain a positive answer for SSS-operators in cotype 2 case (see Theorem 4.12).

(5) In general, for a Banach space Y the validity of the inclusion SS(ℓ_{2}, Y) ⊂ K(ℓ_{2}, Y) may not imply that Y has the Orlicz property. Indeed, let Y be a Banach space with the Schur property, then L(ℓ_{2}, Y) = K(ℓ_{2}, Y) = SS(ℓ_{2}, Y). Take now as Y the ℓ_{1}−sum of spaces ℓ_{∞}^{n}, n = 1, 2, . . .. Then Y has no Orlicz property (this is evident) and has the Schur property (see [5, Corollary 2.4(c)]).

2.4. Diagonalizable operators and their approximation numbers. Fix a family of scalars (λ_{j})_{j∈J}.

Let X be a Banach space with an unconditional basis (x_{j})_{j∈J} and Y be a Banach space with an unconditional basis (y_{j})_{j∈J}. An operator T ∈ L(X, Y) is called diagonal and the family (λ_{j})_{j∈J} is called the diagonal of T with respect to (x_{j})_{j∈J} and (y_{j})_{j∈J} if T x_{j} = λ_{j}y_{j}, ∀j ∈ J.

If X, Y are Banach spaces with unconditional bases (with equal cardinalities), then an operator T ∈ L(X, Y) is called diagonalizable (resp., diagonalizable with the diagonal (λ_{j})_{j∈J}) if there exists an unconditional basis (x_{j})_{j∈J} in X and an unconditional basis (y_{j})_{j∈J} in Y, such that T is diagonal with respect to (x_{j})_{j∈J} and (y_{j})_{j∈J} (resp., with the diagonal (λ_{j})_{j∈J}).

If X is a Banach space with an unconditional basis, then an operator T ∈ L(X, X) is called strictly diagonalizable if there exists an unconditional basis (x_{j})_{j∈J} in X such that T is diagonal with respect to (x_{j})_{j∈J} and (x_{j})_{j∈J}.

If X, Y are Hilbert spaces, then an operator T ∈ L(X, Y) is called orthodiagonalizable if there exists an orthonormal basis (x_{j})_{j∈J} in X and an orthonormal basis (y_{j})_{j∈J} in Y, such that T is diagonal with respect to (x_{j})_{j∈J} and (y_{j})_{j∈J}; an operator T ∈ L(X, X) is called strictly ortho-diagonalizable if there exists an orthonormal basis (x_{j})_{j∈J} in X such that T is diagonal with respect to (x_{j})_{j∈J} and (x_{j})_{j∈J}.

It is well-known that if X, Y are Hilbert spaces, then any compact operator T : X → Y is ortho-diagonalizable, with the diagonal tending to zero (Schmidt’s theorem, see, e.g.,[53, p. 119]).

It is easy to see that if H, Y are Hilbert spaces and T ∈ L(H, Y) is orthodiagonalizable, then T^{∗}T is strictly ortho-diagonalizable and conversely. Notice that if T ∈ L(H) is an ortho-diagonalizable positive self-adjoint operator, then it is strictly ortho-diagonalizable, but we shall not use this further.

Lemma 2.20. Let X be a Banach space with an unconditional basis (e_{j})_{j∈J}, Y be a Banach space, T ∈ L(X, Y) be a non-zero operator, λ_{j} := ∥T e_{j}∥, ∀j ∈ J and J_{+} := {j ∈ J : ∥T e_{j}∥ > 0}.

(a) If (e_{j})_{j∈J} is a Besselian family with a constant b and (\frac{T e_{j}}{∥T e_{j}∥})_{j∈J_{+}} is a Hilbertian family with a constant h , then

a_{n}(T) ≤ b h a_{n}(λ_{·}) ∀n ∈ ℕ.

(b) If (e_{j})_{j∈J} is a Hilbertian family with a constant h and (\frac{T e_{j}}{∥T e_{j}∥})_{j∈J_{+}} is a Besselian family in Y with a constant b, then

a_{n}(T) ≥ b^{−1} h^{−1} a_{n}(λ_{·}) ∀n ∈ ℕ.

Proof. Put y_{j} := \frac{T e_{j}}{∥T e_{j}∥}, ∀j ∈ J_{+} and λ_{·}^{+} := (λ_{j})_{j∈J_{+}}.

(a) Fix a non-empty finite Δ ⊂ J_{+}. Consider the operator T_{Δ} ∈ L(X, Y) defined by the equality:

T_{Δ} x = ∑_{j∈Δ} λ_{j}e_{j}^{∗}(x)y_{j}, x ∈ X.

Since (y_{j})_{j∈J_{+}} is a Hilbertian family with a constant h and (e_{j})_{j∈J_{+}} is a Besselian family with a constant b, we can write:

∥T − T_{Δ}∥ = sup_{∥x∥≤1} ∥ ∑_{j∈J_{+}\Δ} λ_{j}e_{j}^{∗}(x)y_{j}∥ ≤ h sup_{∥x∥≤1} ( ∑_{j∈J_{+}\Δ} |λ_{j}e_{j}^{∗}(x)|^{2})^{\frac{1}{2}}

≤ h sup_{j∈J_{+}\Δ} λ_{j} sup_{∥x∥≤1} (∑_{j∈J} |e_{j}^{∗}(x)|^{2})^{\frac{1}{2}} ≤ h b sup_{j∈J_{+}\Δ} λ_{j}.

Fix now n ∈ ℕ and take a Δ ⊂ J_{+} with card Δ < n. Then since rank T_{Δ} < n, the definition of a_{n}(T) and the above inequality imply:

a_{n}(T) ≤ ∥T − T_{Δ}∥ ≤ h b sup_{j∈J_{+}\Δ} λ_{j}.

Hence,

a_{n}(T) ≤ hb inf_{Δ⊂J_{+}, card Δ<n} sup_{j∈J_{+}\Δ} λ_{j} = h b a_{n}(λ_{·}^{+}) = h b a_{n}(λ_{·})

and (a) is proved.

(b) Fix any Δ ⊂ J_{+} with card Δ = n and denote E the vector subspace of X generated by {e_{j} : j ∈ Δ}. Take now any T_{0} ∈ L(X, Y) with rank T_{0} < n. Since dim(E) = n, the restriction of T_{0} to E cannot be injective, hence ker T_{0} ∩ E ≠ {0}. Then we can find x_{0} ∈ ker T_{0} such that ∥x_{0}∥ = 1 and x_{0} = ∑_{j∈Δ} t_{j}e_{j} for some scalars t_{1}, …, t_{n}. Since (y_{j})_{j∈J_{+}} is a Besselian family with the constant b and (e_{j})_{j∈J} is a Hilbertian family with the constant h, we can write:

∥T − T_{0}∥ ≥ ∥T x_{0} − T_{0}x_{0}∥ = ∥T x_{0}∥ = ∥ ∑_{j∈Δ} t_{j}T e_{j}∥ = ∥ ∑_{j∈Δ} t_{j}λ_{j}y_{j}∥

≥ b^{−1}( ∑_{j∈Δ} |t_{j}λ_{j}|^{2})^{\frac{1}{2}} ≥ b^{−1} min_{j∈Δ} λ_{j}(∑_{j∈Δ} |t_{j}|^{2})^{\frac{1}{2}}

≥ b^{−1} h^{−1} min_{j∈Δ} λ_{j}∥x_{0}∥ = b^{−1} h^{−1} min_{j∈Δ} λ_{j}.

This, as Δ ⊂ J_{+} is arbitrary, gives: ∥T − T_{0}∥ ≥ b^{−1}ã_{n}(λ_{·}^{+}) and, by Lemma 2.2(a,a′), we get: a_{n}(T) ≥ b^{−1} h^{−1}a_{n}(λ_{·}).

Corollary 2.21. If H, Y are Hilbert spaces and T ∈ L(H, Y) is an orthodiagonalizable operator with the diagonal (λ_{j})_{j∈J}, then

a_{n}(T) = a_{n}(λ_{·}) ∀n ∈ ℕ,  (2.5)

and

d(T, K(H, Y)) = u-lim sup_{j∈J} |λ_{j}|.  (2.6)

Proof. Since any orthonormal family is both Hilbertian and Besselian with constant 1, from (a) and (b) we get (2.5). Then by Corollary 2.8(a) and Lemma 2.2(b) we can write

d(T, K(H, Y)) = lim_{n} a_{n}(T) = lim_{n} a_{n}(λ_{·}) = u-lim sup_{j∈J} |λ_{j}|,

i.e., (2.6) holds.

Remark 2.22. (1) Equality (2.5) remains valid also for diagonal operators acting in l_{∞} (see [29, p. 510, Lemma 27.10.3] from where the method of the above given proof is taken partially). The same equality is proved in [52, (11.11.3)] for a diagonal operator T : ℓ_{p} → ℓ_{p} with decreasing nonnegative diagonal.

(2) Let 1 < q < 2 and T : l_{2} → l_{q} be a diagonal operator with respect to natural bases with the diagonal (λ_{n}) such that (|λ_{n}|) is a decreasing sequence. Then since the natural basis of l_{q} is Besselian with constant 1, from Lemma 2.20(a) we get a_{n}(T) ≥ |λ_{n}|, n = 1, 2, . . .. In this case it is known that a_{n}(T) = (∑_{k=n}^{∞} |λ_{k}|^{r})^{\frac{1}{r}} , n = 1, 2, …, where \frac{1}{r} = \frac{1}{q} − \frac{1}{2} [52, (11.11.4)].

3. Inequalities for δ and l-norm

3.1. General inequalities. In this subsection E will denote a non-zero finitedimensional Hilbert space with dim(E) = n over 𝕂 ∈ {ℝ, ℂ}, Y will stand for a general normed space over 𝕂 and T ∈ L(E, Y) will be a fixed operator.

It is well-known that on the Borel σ-algebra of the unit sphere S_{E} of E there exists a unique isometrically invariant probability measure s_{E} called a uniform distribution on S_{E}, which we simply denote by s. This measure is used for the definition of MD-number δ(T) in the introduction (see equality (1.1)).

Clearly E carries the Lebesgue measure λ obtained from any realization of E as 𝕂^{n}.

In what follows, to simplify the notation, we introduce a parameter κ, which equals 1/2 when the considered spaces are real, and equals 1 in the complex case.

Recall now that the standard Gaussian measure γ_{E} on E is a measure such that

γ_{E}(B) = (\frac{κ}{π} )^{κ n} ∫_{B} exp(−κ∥x∥_{E}^{2})dλ(x)

for any Borel subset B of E.

Observe that γ_{E} is also isometrically invariant Borel probability measure on E, which is also uniquely defined through its Fourier transform

γ̂_{E}(h) := ∫_{E} exp{i Re (x|h)} d γ_{E}(x) = exp{−κ∥h∥_{E}^{2}} ∀ h ∈ E. (3.1)

Let

a_{p}(γ_{E}) := (∫_{E} ∥x∥^{p} d γ_{E}(x))^{\frac{1}{p}} , 0 < p < ∞.  (3.2)

The exact value of a_{p}(γ_{E}) will not be important in the sequel. It is easy to see that

a_{p}(γ_{E}) ≤ a_{2}(γ_{E}) = \sqrt{dim(E)} ∀p ∈]0, 2]  (3.3)

while,

a_{1}(γ_{ℝ}) = \sqrt{\frac{2}{π}} , a_{1}(γ_{ℂ}) = \frac{\sqrt{π}}{2} .  (3.4)

The following statement relates γ_{E} with s_{E} and is not difficult to check (using the uniqueness of s_{E}): if f : E → ℂ is a positively homogeneous measurabe function, then

(∫_{E} |f(x)|^{p} d γ_{E}(x))^{\frac{1}{p}} = a_{p}(γ_{E}) (∫_{S_{E}} |f(x)|^{p} d s_{E}(x))^{\frac{1}{p}} , 0 < p < ∞. (3.5)

The measure γ_{E} is used for the definition of l(T) in the Introduction (see equality (1.3)).

We shall use frequently formula (1.2)(see Introduction) which follows at once from equality in (3.3) and (3.5) applied for p = 2 to x → f(x) := ∥T x∥.

Evidently, we have

m(T) ≤ δ(T) ≤ ∥T ∥, ∀T ∈ L(E, Y)

where m(T) is the lower bound of T.

To get some other more useful observations, some facts about Gaussian measures and Gaussian random variables are necessary.

Fix a probability space (Ω, A, ℙ) and a sequence (g_{k})_{k∈ℕ} of independent standard 𝕂-valued Gaussian random variables on this space.^{11} We denote by 𝔼 the integral with respect to probability measure ℙ.

Theorem 3.1 ([37, p. 1925, Corollary 5]). Let Y be a normed space over 𝕂, (g_{k})_{k∈ℕ} be a sequence of independent standard 𝕂-valued random variables, n ∈ ℕ, y_{1}, y_{2}, …, y_{n} ∈ Y and 0 < r < p < ∞. Then we have

(𝔼∥ ∑_{k=1}^{n} g_{k}y_{k}∥^{p})^{\frac{1}{p}} ≤ K_{p,r} (𝔼∥ ∑_{k=1}^{n} g_{k}y_{k}∥^{r})^{\frac{1}{r}}

with the constant

K_{p,r} = \frac{a_{p}(γ_{ℝ})}{a_{r}(γ_{ℝ})},

which is the best possible in the real case. In particular,

(𝔼∥ ∑_{k=1}^{n} g_{k}y_{k}∥^{2})^{\frac{1}{2}} ≤ \sqrt{\frac{π}{2}} 𝔼∥ ∑_{k=1}^{n} g_{k}y_{k}∥.

This theorem with some universal constant C_{p,r} depending only on p and r, was well-known earlier^{12}. We also note that in [37] this statement is proved in the real case; the complex case (with the same K_{p,r} as in the real case) follows easily from this.

Lemma 3.2. Let E be a finite-dimensional Hilbert space with dim E = n ≥ 1, Y be a normed space, T ∈ L(E, Y) and e_{k}, k = 1, …, n be any fixed orthonormal basis of E. Then γ_{E} coincides with the distribution^{13} of the mapping ω → ∑_{k=1}^{n} g_{k}(ω)e_{k} in E and

l(T) = (𝔼∥ ∑_{k=1}^{n} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} .  (3.6)

11 This means that the random variables g_{k} : Ω → 𝕂, k = 1, 2, …, are stochastically independent and g_{k} has the distribution γ_{𝕂} for any k ∈ ℕ. E.g., it is possible to take Ω = 𝕂^{ℕ}, ℙ = ∏_{k∈ℕ} μ_{k}, where μ_{k} = γ_{𝕂}, k = 1, 2, …, and (g_{k}) is the sequence of coordinate functionals.

12 See, e.g., [62, Theorem 5.5.3(d)]; the proof of [55, Corollary 4.9, p.48] gives the estimation C_{2,1} ≤ π\sqrt{\frac{π}{2}} .

13 i.e., with the image of ℙ.

Moreover, we have

∫_{E} ∥T x∥dγ_{E}(x) ≤ l(T) ≤ \sqrt{\frac{π}{2}} ∫_{E} ∥T x∥dγ_{E}(x)  (3.7)

and

∫_{S_{E}} ∥T x∥ds_{E}(x) ≤ δ(T) ≤ \sqrt{\frac{π}{2}} ∫_{S_{E}} ∥T x∥ds_{E}(x).  (3.8)

Proof. Denote by μ the image of ℙ under the mapping ω → ∑_{k=1}^{n} g_{k}(ω)e_{k}. Then μ is a probability measure on E. It is easy to observe (using the independence of g_{k}, k = 1, …, n and equality (3.1)) that μ̂ = γ̂_{E}. Hence μ = γ_{E} (in view of the uniqueness theorem for the Fourier transform).

Equality (3.6) now follows from the definition of l and from the change of variable formula.

The left inequalities in (3.7) and (3.8) are evident. The right inequality in (3.7) follows from (3.6) and Theorem 3.1. Applying then (1.2), (3.7), (3.5)(for p = 1) and (3.3), we get the relation

\sqrt{n} δ(T) = l(T) ≤ \sqrt{\frac{π}{2}} ∫_{E} ∥T x∥dγ_{E}(x)

= \sqrt{\frac{π}{2}} a_{1}(γ_{E}) ∫_{S_{E}} ∥T x∥ds_{E}(x) ≤ \sqrt{\frac{π}{2}} \sqrt{n} ∫_{S_{E}} ∥T x∥ds_{E}(x),

which gives the right inequality in (3.8).

Relation (3.8) shows that for a given operator T its MD-number δ(T) and its “Levy’s mean number” ∫_{S_{E}} ∥T x∥ds_{E}(x) are equivalent quantities (the latter quantity is used, e.g., in [45]).

Corollary 3.3. We have

∥T e∥ ≤ l(T), \frac{1}{\sqrt{dim E}} ∥T e∥ ≤ δ(T) ∀e ∈ S_{E}  (3.9)

Moreover, the functionals l and δ are norms on L(E, Y) with properties

∥T ∥ ≤ l(T) ≤ \sqrt{dim E} ∥T ∥,  (3.10)
\frac{1}{\sqrt{dim E}} ∥T ∥ ≤ δ(T) ≤ ∥T ∥.  (3.11)

Proof. Let e_{k}, k = 1, …, n be any orthonormal basis of E. Then via (3.6)

∥T e_{1}∥ ≤ l(T) ≤ ∥T ∥ (𝔼 ∥∑_{k=1}^{n} g_{k}e_{k}∥_{E}^{2})^{\frac{1}{2}} = \sqrt{n}∥T ∥.

Since any element of S_{E} can be taken as e_{1}, this relation implies (3.10). Evidently, (3.10) and (1.2) imply (3.11). It is clear that the functionals l and δ are seminorms on L(E, Y). From (3.10) and (3.11) we get that they actually are norms equivalent to the usual operator norm.

Corollary 3.4. Let M ⊂ E be a fixed non-zero vector subspace. Then:

l(T |_{M}) ≤ l(T),  (3.12)
δ(T |_{M}) ≤ \sqrt{\frac{dim E}{dim M}} δ(T).  (3.13)

Moreover, if M_{1}, M_{2} ⊂ E are any (i.e., not necessarily algebraically complementary to each other) vector subspaces with M_{1} + M_{2} = E, then we have also

δ(T) ≥ \frac{δ(T |_{M_{2}})}{\sqrt{\frac{dim M_{1}}{dim M_{2}} + 1}},  (3.14)

δ(T) ≤ \sqrt{\frac{dim M_{1}}{dim M_{1} + dim M_{2}}} ∥T ∥ + δ(T |_{M_{2}}).  (3.15)

Proof. Let dim M = m. We can suppose that the basis e_{k}, k = 1, …, n is chosen in such a way that e_{k}, k = 1, …, m is an orthonormal basis of M. Now (3.12) follows from (3.6) via the inequality

(𝔼 ∥∑_{k=1}^{m} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} ≤ (𝔼 ∥∑_{k=1}^{n} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} ,

which is in turn an easy consequence of the independence and symmetry of g_{k}, k = 1, …, n (see, e.g., [62, Lemma 5.3.4(a), p.281]).

Inequality (3.13) follows from (3.12) and (1.2).

Using (1.2) and (3.12) for M = M_{2}, we can write δ(T) ≥ \frac{ℓ(T |_{M_{2}})}{\sqrt{dim E}}. Since dim E ≤ dim M_{1} + dim M_{2}, from this we get (3.14).

The inequality (3.15) needs a little more work (and linear algebra). Denote byM′ the orthogonal complement of M_{2} into E and n′ = dim M′. We can suppose now that the basis e_{k}, k = 1, …, n is chosen in such a way that e_{k}, k = 1, …, n′ is an orthonormal basis of M′. Then using the subadditivity of the 𝕃_{2}(Ω, A, ℙ) norm and (3.6) we get

l(T) ≤ (𝔼 ∥∑_{k=1}^{n′} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} + (𝔼 ∥ ∑_{k=n′+1}^{n} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} = l(T |_{M′}) + l(T |_{M_{2}}).

This and (1.2) give

δ(T) ≤ \sqrt{\frac{n′}{n}} δ(T |_{M′}) + \sqrt{\frac{dim(M_{2})}{n}}δ(T |_{M_{2}}) ≤ \sqrt{\frac{n′}{n}} δ(T |_{M′}) + δ(T |_{M_{2}}).

Since by (3.11) we have δ(T |_{M′}) ≤ ∥T |_{M′}∥ ≤ ∥T ∥ and n′ ≤ dim(M_{1}) (as n′ + dim(M_{2}) = dim(M) ≤ dim(M_{1}) + dim(M_{2}), hence n′ ≤ dim(M_{1})), we get

δ(T) ≤ \sqrt{\frac{n′}{n′ + dim M_{2}}} ∥T ∥ + δ(T |_{M_{2}}) ≤ \sqrt{\frac{dim(M_{1})}{dim(M_{1}) + dim(M_{2})}}∥T ∥ + δ(T |_{M_{2}}),

i.e., (3.15) holds.

The next statement deals with the ideal properties of the norms l and δ.

Proposition 3.5. Let E be a finite-dimensional Hilbert space with dim E ≥ 1, Y be a normed space and T ∈ L(E, Y). Suppose further that E_{1} is another finite-dimensional Hilbert space with dim E_{1} ≥ 1, Y_{1} another normed space, u ∈ L(E_{1}, E) and v ∈ L(Y, Y_{1}). Then:

l(vT) ≤ ∥v∥l(T), δ(vT) ≤ ∥v∥δ(T),  (3.16)
l(T u) ≤ ∥u∥l(T)  (3.17)

and

δ(T u) ≤ \sqrt{\frac{dim E}{dim E_{1}}} δ(T)∥u∥.  (3.18)

Proof. The inequality l(vT) ≤ ∥v∥l(T) is evident. The inequality l(T u) ≤ l(T)∥u∥ can be shown as follows. Let n := dim E and n_{1} := dim E_{1}. We have

ux = ∑_{k=1}^{n_{1}} λ_{k}(x|e′_{k})e_{k}, x ∈ E_{1},

where e′_{k}, k = 1, …, n_{1} is an orthonormal basis of E_{1}, e_{k}, k = 1, …, n_{1} is a finite sequence in E whose non-zero members are orthonormal and λ_{k}, k = 1, …, n_{1} are non-negative numbers such that ∥u∥ = λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n_{1}}. Then

T ux = ∑_{k=1}^{n_{1}} λ_{k}(x|e′_{k})T e_{k}, ∀x ∈ H_{1}.

Using this equality, (3.6) and the contraction principle (see [62, Lemma 5.4.1(c), p.298]), we get

l(T u) = (𝔼 ∥∑_{k=1}^{n_{1}} g_{k}λ_{k}T e_{k}∥^{2})^{\frac{1}{2}} ≤ ( max_{1≤k≤n_{1}} λ_{k}) (𝔼 ∥∑_{k=1}^{n_{1}} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} ≤ l(T)∥u∥.

(3.18) follows from (3.17) and (1.2).

3.2. Inequalities involving medians. Let μ be a probability measure given on a σ-algebra A of subsets of a set Ω and f : Ω → ℝ be a A-measurable function. There exists at least one t ∈ ℝ such that

μ\{ω ∈ Ω : f(ω) ≥ t\} ≥ \frac{1}{2} ≤ μ\{ω ∈ Ω : f(ω) ≤ t\}.  (3.19)

The set M_{f,μ} of all t ∈ ℝ for which (3.19) holds either consists of one single point or is a nondegenerate closed segment. Any member of M_{f,μ} is called a median of f with respect to μ.

Lemma 3.6. Let (Ω, A, μ) be probability space and f : Ω → ℝ be a Ameasurable function.

(a) If f ≥ 0 μ-a.e., f ∈ 𝕃_{2}(Ω, A, μ), ∥f∥_{2} > 0 and α := \frac{∥f∥_{1}}{∥f∥_{2}} , then

(α − \sqrt{1 − α^{2}})∥f∥_{2} ≤ t ≤ \sqrt{2}∥f∥_{2} ∀t ∈ M_{f,μ}.  (3.20)

Moreover, if B ⊂ Ω is a measurable set such that μ(B) = \frac{1}{2} and f := 1_{B}, then Med_{μ}(f) = [0, 1] and in (3.20) we have equalities for some medians.

(b) If Ω is a connected topological space, μ is a Borel probability measure, which is strictly positive on any non-void open subset of Ω and f is continuous, then the median of f is unique.

Proof. (a) Fix t ∈ M_{f,μ}. An application of the (Markov) inequality

μ{ω ∈ Ω : f(ω) ≥ r} ≤ r^{−2}∥f∥_{2}, r > 0,

for r = \sqrt{2}∥f∥_{2} gives: μ\{ω ∈ Ω : f(ω) ≥ \sqrt{2}∥f∥_{2}\} ≤ \frac{1}{2}. Hence t ≤ \sqrt{2}∥f∥_{2}.

The left estimation in (3.20) is derived from the following remarkable known property of a median:

∥f − t∥_{1} ≤ ∥f − x∥_{1}, ∀ x ∈ ℝ.

For x = ∥f∥_{1} this inequality gives: t ≥ ∥f∥_{1} − ∥f − ∥f∥_{1}∥_{1}. Since

∥f − ∥f∥_{1}∥_{1} ≤ ∥f − ∥f∥_{1}∥_{2} = \sqrt{∥f∥_{2}^{2} − ∥f∥_{1}^{2}},

we get

t ≥ ∥f∥_{1} − ∥f − ∥f∥_{1}∥_{1} ≥ ∥f∥_{1} − \sqrt{∥f∥_{2}^{2} − ∥f∥_{1}^{2}} = (α − \sqrt{1 − α^{2}})∥f∥_{2}.

The rest is evident.

(b) Suppose that there are t_{1}, t_{2} ∈M_{f,μ} such that t_{1} <t_{2}. Then μ (f^{−1}]t_{1}, t_{2}[)= 0. Since t_{1}, t_{2} are medians of f, we can find real numbers r_{1}, r_{2} ∈ f(Ω) such that r_{1} ≤ t_{1} and r_{2} ≥ t_{2}. Since Ω is connected and f is continuous, f(Ω) is a connected subset of ℝ. Hence f(Ω) is convex and then [r_{1}, r_{2}] ⊂ f(Ω). In particular, [t_{1}, t_{2}] ⊂ f(Ω). This implies that f^{−1}(]t_{1}, t_{2}[) is a non-void open set and because of the required property of μ, it must have a strictly positive measure. A contradiction.

Let T ∈ L(E, Y). Consider the function f_{T} : E → ℝ_{+} defined by the equality f_{T} (x) = ∥T x∥, x ∈ E. By Lemma 3.6(b) the function f_{T} |_{S_{E}} has a unique median with respect to the uniform distribution s = s_{E}, which we denote by Med_{s}(T) and call the median of T with respect to s. Analogously, f_{T} has a unique median with respect to the standard Gaussian measure γ_{E}, which we denote by Med_{γ}(T) and call it the median of T with respect to γ.

Proposition 3.7. Let E be a finite-dimensional Hilbert space with dim E = n ≥ 1, Y be a normed space and T ∈ L(E, Y). Then we have

\frac{1}{3\sqrt{π}} δ(T) ≤ \frac{\sqrt{2} − \sqrt{π − 2}}{\sqrt{π}} δ(T) ≤ Med_{s}(T) ≤ \sqrt{2} δ(T)  (3.21)

and

\frac{1}{3\sqrt{π}} l(T) ≤ \frac{\sqrt{2} − \sqrt{π − 2}}{\sqrt{π}} l(T) ≤ Med_{γ}(T) ≤ ∫_{E} ∥T x∥ d γ_{E}(x) ≤ l(T). (3.22)

Proof. Inequalities (3.21) and the left-hand side of (3.21) follow from Lemma 3.6 applied to f = f_{T} |_{S_{E}} in the first case and to f = f_{T} in the second case and taking into account that in either cases, according to Lemma 3.2 we have the inequality α = \frac{∥f∥_{1}}{∥f∥_{2}} ≥ \sqrt{\frac{2}{π}}. The inequality

Med_{γ}(T) ≤ ∫_{E} ∥T x∥ d γ_{E}(x)

is a particular case of a general result of [34], which states that the same is true for any Gaussian measure and any convex functional.

Remark 3.8. (1) This proposition shows that for a given operator T the median Med_{s}(T) and the MD-number δ(T) are equivalent quantities.

(2) Proposition 3.7 for Med_{s}(T) coincides with [61, p. 26, Lemma 7.2], where it is proved in a different way for some numerical constant in place of \frac{1}{3\sqrt{π}} (of course, without having the exact estimate (3.8)).

The next assertion demonstrates that the evident inequalities m(T) ≤ δ(T) and δ(T) ≤ ∥T ∥ can be inverted. The proof involves in fact the Isoperimetric Inequality.

Proposition 3.9. There exists universal constant C such that the following is true:

Let E be a finite-dimensional Hilbert space with dim E ≥ 1, Y be a normed space, T ∈ L(E, Y) and τ ∈ [∥T ∥, ∞[. Then for each ε ∈]0, \frac{τ}{2}[ there exists a vector subspace F ⊂ E with

dim(F) ≥ C \frac{ε^{2} dim(E)}{| log \frac{ε}{τ} |τ^{2}},  (3.23)

such that

δ(T) ≤ 3\sqrt{π} (m(T |_{F}) + ε)  (3.24)

and

\sqrt{2}(∥T |_{F} ∥ − ε) ≤ δ(T).  (3.25)

Proof. Let f(x) = ∥T x∥, x ∈ S_{E}. Then f is a τ-Lipschitz function on S_{E} with respect to the Euclidean norm and its median coincides with Med_{s}(T). An application of fundamental Proposition 12.3 from [2, p. 284] to f gives the existence of F satisfying (3.23) and having the property

Med_{s}(T) + ε ≥ ∥T x∥ = f(x) ≥ Med_{s}(T) − ε, ∀x ∈ S_{F} .

Hence

m(T |_{F}) := inf{∥T x∥ : x ∈ S_{F} } ≥ Med_{s}(T) − ε  (3.26)

and

Med_{s}(T) + ε ≥ sup{∥T x∥ : x ∈ S_{F} } := ∥T |_{F} ∥.  (3.27)

Now, (3.26) and Proposition 3.7 imply (3.24); analogously, (3.27) and Proposition 3.7 imply (3.25).

3.3. Inequalities related with type and cotype 2. In Section 2 we have already used the notions of Rademacher type 2 and Rademacher cotype 2 Banach spaces. In what follows it would be more convenient to deal with the “Gaussian” versions of these notions.

A non-trivial normed space Y is called:

• of Gaussian type 2 if there is a constant c > 0 such that for each n ∈ ℕ and y_{1}, y_{2}, …, y_{n} ∈ Y,

(𝔼∥ ∑_{k=1}^{n} g_{k}y_{k}∥^{2})^{\frac{1}{2}} ≤ c (∑_{k=1}^{n} ∥y_{k}∥^{2})^{\frac{1}{2}} .  (3.28)

If Y is of Gaussian type 2, then the best possible constant c for which (3.28) holds is called the Gaussian type 2 constant of Y.

• of Gaussian cotype 2 if there is a constant c > 0 such that for each n ∈ ℕ and y_{1}, y_{2}, …, y_{n} ∈ Y,

c (𝔼∥ ∑_{k=1}^{n} g_{k}y_{k}∥^{2})^{\frac{1}{2}} ≥ (∑_{k=1}^{n} ∥y_{k}∥^{2})^{\frac{1}{2}} .  (3.29)

If Y is of Gaussian cotype 2, then the best possible constant c for which (3.28) holds is called the Gaussian cotype 2 constant of Y.

It is known and it is not hard to show that the Gaussian type 2 and Rademacher type 2 are equivalent notions (see, e.g.,[18, Theorem 12.26]). It is also known and it is a highly delicate result that the Gaussian cotype 2 and Rademacher cotype 2 are equivalent notions (see, [18, Corollary 12.28] and the text following it). If Y is an inner-product space, then Y is both of type 2 and of cotype 2 with constant one (this is easy to see). The converse is also true [33]: if a normed space Y is simultaneously of type 2 and cotype 2, then Y is isomorphic to an inner-product space.

Remark 3.10. Let Y be a non-trivial normed space.

(a) Y is of Gaussian type 2, if and only if there is a constant c > 0 such that for any finite-dimensional Hilbert space E, n := dim E ≥ 1, any orthonormal basis e_{1}, …, e_{n} and any T ∈ L(E, Y) we have the inequality

l(T) ≤ c (∑_{k=1}^{n} ∥T e_{k}∥^{2})^{1/2} .  (3.30)

(b) Y is of Gaussian cotype 2, if and only if there is a constant c > 0 such that for any finite-dimensional Hilbert space E, n := dim(M), any orthonormal basis e_{1}, …, e_{n} and any T ∈ L(E, Y) we have the inequality

l(T) ≥ c^{−1} (∑_{k=1}^{n} ∥T e_{k}∥^{2})^{1/2} .  (3.31)

(c) It is worthwhile to note that without any further supposition about Y, the following is true: for any finite-dimensional Hilbert space E, n := dim E, any T ∈ L(E, Y) there are orthonormal bases e′_{1}, …, e′_{n} and e′′_{1}, …, e′′_{n} of E such that we have the inequality

(∑_{k=1}^{n} ∥T e′_{k}∥^{2})^{1/2} ≤ l(T) ≤ (∑_{k=1}^{n} ∥T e′′_{k}∥^{2})^{1/2} .

This fact we shall not use in what follows, its proof in fact is given in [35]. From Remark 3.10(a), (b) we get the following statement.

Lemma 3.11. Let E be a finite-dimensional Hilbert space with dim E = n ≥ 1, Y a normed space, T ∈ L(E, Y) and (e_{1}, …, e_{n}) any fixed orthonormal basis of E. Then:

(a) If Y is of Gaussian cotype 2, then

δ(T) ≥ c^{−1} min_{k≤n} ∥T e_{k}∥,  (3.32)

where c is the cotype 2 constant of Y.

(b) If Y is of Gaussian type 2, then

δ(T) ≤ c max_{k≤n} ∥T e_{k}∥,  (3.33)

where c is the type 2 constant of Y.

Proof. (a) Taking into account (1.2), (3.6) and (3.29), we get

\sqrt{n}δ(T) = l(T) = (𝔼 ∥∑_{k=1}^{n} g_{k}T e_{k}∥^{2})^{\frac{1}{2}} ≥ c^{−1} (∑_{k=1}^{n} ∥T e_{k}∥^{2})^{1/2}

≥ c^{−1} \sqrt{n} min_{k≤n} ∥T e_{k}∥.

(b) follows in a similar way from (1.2), (3.6) and (3.28).

4. General Properties of AMD-Numbers

4.1. Continuity properties of AMD-numbers. Let now H be an infinite dimensional Hilbert space and Y be a normed space. In this case for a given operator T ∈ L(H, Y) we have already defined its upper AMD-number \overline{\overline{δ}}(T), the lower AMD-number \underline{\underline{δ}}(T) and the notion of an AMD-regular operator in the Introduction.

It is possible to rewrite these definitions in a more direct form without using upper and lower limits. For this fix an M ∈ FD(H) and put

β_{M}(T) := sup{δ(T |_{N}) : N ⊃ M, N ∈ FD(H)}

and

α_{M}(T) := inf{δ(T |_{N}) : N ⊃ M, N ∈ FD(H)}.

Using these notations we get:

\overline{\overline{δ}}(T) = inf_{M∈FD(H)} β_{M}(T)  (4.1)

and

\underline{\underline{δ}}(T) = sup_{M∈FD(H)} α_{M}(T).  (4.2)

If \overline{\overline{δ}}(T) = \underline{\underline{δ}}(T), then T is named AMD-regular. Clearly, for an AMD-regular T the net (δ(T |_{M}))_{M∈FD} is convergent and \overline{δ}(T) := lim_{M∈FD} δ(T |_{M}). Let us denote by AMDR(H, Y) the subset of all AMD-regular operators from L(H, Y). This is a proper subset of L(H, Y) (provided H and Y are infinite-dimensional Hilbert spaces, see Remark 4.5).

The next assertion collects the continuity properties of asymptotic mean dilatation numbers.

Proposition 4.1. Let Y be a normed space. Then:

(a) T → \overline{\overline{δ}}(T) is a continuous seminorm on the normed space L(H, Y) with property \overline{\overline{δ}}(T) ≤ ∥T ∥, ∀ T ∈ L(H, Y).

In particular, ker(\overline{\overline{δ}}) := \{T ∈ L(H, Y) : \overline{\overline{δ}}(T) = 0\} is a closed vector subspace of L(H, Y).

(a′) If Y_{1} is a normed space, then

\overline{\overline{δ}}(vT) ≤ ∥v∥\overline{\overline{δ}}(T) ∀ T ∈ L(H, Y) ∀ v ∈ L(Y, Y_{1}).  (4.3)

In particular, ker(\overline{\overline{δ}}) is a left ideal.

(b) \underline{\underline{δ}} has property

|\underline{\underline{δ}}(T_{1}) − \underline{\underline{δ}}(T_{2})| ≤ ∥T_{1} − T_{2}∥ ∀ T_{1}, T_{2} ∈ L(H, Y).

In particular, \underline{\underline{δ}} is continuous on L(H, Y) and ker(\underline{\underline{δ}}) is a closed subset of L(H, Y) (which is not a vector subspace when Y = H, see Remark 4.5).

(c) AMDR(H, Y) is a closed subset in L(H, Y) (AMDR(H, H) is not a vector subspace of L(H, H), see Remark 4.5) and δ̄ is a continuous functional on it.

Proof. (a) follows easily from the fact that for any fixed M ∈ FD the functional δ is a norm on L(M, Y) and from the evident inequality δ(T |_{M}) ≤ ∥T |_{M}∥ ≤ ∥T ∥.

(b) Fix T_{1}, T_{2} ∈ L(H, Y) and a finite dimensional M ⊂ H. We can write

δ(T_{2}|_{M}) ≤ δ((T_{2} − T_{1})|_{M}) + δ(T_{1}|_{M}) ≤ ∥T_{2} − T_{1}∥ + δ(T_{1}|_{M}).

This inequality implies at once \underline{\underline{δ}}_{1}(T_{2}) ≤ ∥T_{2} − T_{1}∥ + \underline{\underline{δ}}_{1}(T_{1}). Consequently, \underline{\underline{δ}}_{1}(T_{2}) − \underline{\underline{δ}}_{1}(T_{1}) ≤ ∥T_{2} − T_{1}∥. Similarly, we have \underline{\underline{δ}}_{1}(T_{1}) − \underline{\underline{δ}}_{2}(T_{1}) ≤ ∥T_{2} − T_{1}∥. Hence we have (b).

(c) follows from (a) and (b).

The next statement provides, in particular, a “sequential” way for computation of AMD-mumbers.

Proposition 4.2. Let Y be a normed space and T ∈ L(H, Y).

(a) We have:

\overline{\overline{δ}}(T) = sup_{(E_{n})} lim sup_{n} δ(T |_{E_{n}})  (4.4)

and

\underline{\underline{δ}}(T) = inf_{(E_{n})} lim inf_{n}δ(T |_{E_{n}}),  (4.5)

where supremum in (4.4), resp., infimum in (4.5) is taken with respect to all sequences (E_{n})_{n∈ℕ} of non-trivial finite-dimensional subspaces of H such that

lim_{n} dim E_{n} = ∞.

(b) If T is AMD-regular and (E_{n})_{n∈ℕ} an arbitrary sequence of non-trivial finite-dimensional subspaces of H such that lim_{n} dim E_{n} = ∞, then lim_{n} δ(T |_{E_{n}}) exists and \overline{δ}(T) = lim_{n} δ(T |_{E_{n}}).

Proof. Let (E_{n})_{n∈ℕ} be any sequence of non-trivial finite-dimensional subspaces of H such that lim_{n} dim E_{n} = ∞.

(a) (1) Let us verify that

\overline{\overline{δ}}(T) ≥ lim sup_{n} δ(T |_{E_{n}}).  (4.6)

Let M_{0} be any finite-dimensional subspace of H. Fix a natural n and put M_{n} = M_{0} + E_{n}. We can write using (3.14) (for E = M_{n}, M_{2} = E_{n}, M_{0} = M_{1})

β_{M_{0}}(T) ≥ δ(T |_{M_{n}}) ≥ \frac{δ(T |_{E_{n}}}{\sqrt{\frac{dim M_{0}}{dim E_{n}} + 1}}.  (4.7)

Since n is arbitrary, this inequality implies β_{M_{0}}(T)≥lim sup_{n} δ(T |_{E_{n}}). Since M_{0} is also arbitrary, we get (4.6). The inequality \overline{\overline{δ}}(T)≤sup_{(E_{n})} lim sup_{n} δ(T |_{E_{n}}) is a direct consequence of the definition of \overline{\overline{δ}}(T).

(2) Let us verify now that

\underline{\underline{δ}}(T) ≤ lim inf_{n}δ(T |_{E_{n}}).  (4.8)

Let M_{0} be any finite-dimensional subspace of H. Fix a natural n and put M_{n} = M_{0} + E_{n}. We can write using (3.15):

α_{M_{0}}(T) ≤ δ(T |_{M_{n}}) ≤ \sqrt{\frac{dim M_{0}}{dim M_{0} + dim E_{n}}} ∥T |_{M_{n}}∥ + δ(T |_{E_{n}})

≤ \sqrt{\frac{dim M_{0}}{dim M_{0} + dim E_{n}}} ∥T |∥ + δ(T |_{E_{n}}).

Since n is arbitrary, from this inequality we obtain α_{M_{0}}(T) ≤ lim inf_{n} δ(T |_{E_{n}}). Since M_{0} is also arbitrary, we get (4.8). The inequality

\underline{\underline{δ}}(T) ≥ inf_{(E_{n})} lim inf_{n}δ(T |_{E_{n}})

is a direct consequence of the definition of \underline{\underline{δ}}(T).

(b) follows from (a).

Remark 4.3. Let H be a separable Hilbert space, (e_{n}) be an orthonormal basis of H and for a fixed n, E_{n} be the vector subspace generated by {e_{1}, …, e_{n}}. One can expect that in this case for any T ∈ L(H) the equality \overline{\overline{δ}}(T) = lim sup_{n} δ(T |_{E_{n}}) should be true. But this is not so. Using Proposition 4.7(a) it is easy to see that for the positive diagonal operator T from Remark 4.5(1) we have

\overline{\overline{δ}}(T) = 1 > \frac{1}{\sqrt{2}} = lim sup_{n} δ(T |_{E_{n}}).

It is not clear whether there exists a “universal” sequence (E_{n}) which could be used for the exact computation of \overline{\overline{δ}}(T) for all possible operators T.

4.2. Orthonormal sequences and AMD-number. In this subsection we show that in certain cases AMD-numbers control and are controlled by the behaviour of an operator on orthonormal sequences.

Proposition 4.4. Let Y be a normed space, T ∈ L(H, Y) and (e_{n}) any infinite orthonormal sequence in H. The following assertions are valid:

(a) If Y is of cotype 2, then

\overline{\overline{δ}}(T) ≥ \frac{1}{c} lim sup_{n} ∥T e_{n}∥,

where c is the cotype-2 constant of Y.

(b) If Y is of type 2, then

\underline{\underline{δ}}(T) ≤ c lim inf_{n} ∥T e_{n}∥,

where c is the type-2 constant of Y.

(c) If Y is an inner-product space and T is AMD-regular, then lim_{n} ∥T e_{n}∥ exists and \overline{δ}(T) = lim_{n} ∥T e_{n}∥.

Proof. (a) Put lim sup_{n} ∥T e_{n}∥ = λ. We can assume λ > 0. There is a subsequence (∥T e_{k_{n}}∥) such that

lim_{n} ∥T e_{k_{n}}∥ = λ.

Fix ε, 0 < ε < λ. Then there is j_{ε} ∈ ℕ such that

∥T e_{k_{j}}∥ > λ − ε ∀j > j_{ε}.

Fix now n ∈ ℕ arbitrarily and consider the vector subspace E_{n} of H generated by {e_{k_{jε+1}}, …, e_{k_{jε+n}}}. Since Y is of cotype 2, we can apply (3.32) for E = E_{n} and T |_{E_{n}} and write

δ(T |_{E_{n}}) ≥ c^{−1} min_{j_{ε}<j≤n} ∥T e_{k_{j}}∥ > c^{−1}(λ − ε).

Since n is arbitrary, this inequality together with Proposition 4.2(a) implies \overline{\overline{δ}}(T) ≥ c^{−1}(λ − ε). But ε > 0 is arbitrary. Consequently, \overline{\overline{δ}}(T) ≥ \frac{1}{c}λ and (a) is proved.

(b) Put lim inf_{n} ∥T e_{n}∥ = λ. There is a subsequence (∥T e_{k_{n}}∥) such that

lim_{n} ∥T e_{k_{n}}∥ = λ.

Fix ε, 0 < ε. Then there is j_{ε} ∈ ℕ such that

∥T e_{k_{j}}∥ < λ + ε ∀j > j_{ε}.

Fix now n ∈ ℕ arbitrarily and consider the vector subspace E_{n} of H generated by {e_{k_{jε+1}}, …, e_{k_{jε+n}}}. Since Y is of type 2, we can apply (3.33) for E = E_{n} and T |_{E_{n}} and write

δ(T |_{E_{n}}) ≤ c max_{j_{ε}<j≤n} ∥T e_{k_{j}}∥ < c(λ + ε).

Since n is arbitrary, this inequality together with Proposition 4.2(a) implies \underline{\underline{δ}}(T) ≤ c(λ + ε). But ε > 0 is arbitrary. Consequently \underline{\underline{δ}}(T) ≤ cλ and (b) is proved.

(c) follows from (a) and (b) since any inner-product space is both of type 2 and of cotype 2 with constant 1.

Remark 4.5. Using Proposition 4.4 it is easy to provide several examples. (1)An example of an operator T which is not AMD-regular. Let H be an

infinite-dimensional separable Hilbert space with orthonormal basis {e_{n} : n ∈ ℕ}, consider the operator T : H → H defined: T e_{1} = e_{1}; T e_{2} = 0; T e_{3} = e_{3}; T e_{4} = 0 . . .. Then for such an operator \overline{\overline{δ}}(T) = 1 (by Proposition 4.4(a) and \underline{\underline{δ}}(T) = 0 (by Proposition 4.4(b)). Hence T is not AMD-regular.

(2) AMDR (H, H) is not a vector space. Evidently, for any isometric operator T : H → H we have \overline{δ}(T) = 1. However there are two unitary operators T_{1} and T_{2} such that \overline{δ}(T_{1} + T_{2}) does not exist. To provide an example, take T_{1} = I identity operator, and define T_{2} by the relations T_{2}(e_{2n}) = e_{2n} and T_{2}(e_{2n−1}) = −e_{2n−1} ∀n. Clearly, T_{2} is a unitary operator but (T_{1} + T_{2})(e_{2n}) = 2e_{2n} and (T_{1} + T_{2})(e_{2n−1}) = 0 for all n. According to Proposition 4.4(a,b), we have \overline{\overline{δ}}(T_{1} + T_{2}) = 2 and \underline{\underline{δ}}(T_{1} + T_{2}) = 0.

(3) ker(\underline{\underline{δ}}) is not a vector space. The same idea as in (2).

The next assertion will allow us to compute exactly AMD-numbers for the diagonalizable operators.

Proposition 4.6. Let Y be a Banach space, T ∈ L(H, Y) be a non-zero operator and (e_{j})_{j∈J} an orthonormal basis in H. Put J_{+} := {j ∈ J : ∥T e_{j}∥ > 0}.

(a) If (\frac{T e_{j}}{∥T e_{j}∥})_{j∈J_{+}} is a Hilbertian family in Y with a constant c, then

\overline{\overline{δ}}(T) ≤ c u-lim sup_{j∈J} ∥T e_{j}∥.

(b) If (\frac{T e_{j}}{∥T e_{j}∥})_{j∈J_{+}} is a Besselian family in Y with a constant c , then

\underline{\underline{δ}}(T) ≥ c^{−1} u-lim inf_{j∈J} ∥T e_{j}∥.

Proof. (a) Since u-lim sup_{j∈J} ∥T e_{j}∥ = u-lim sup_{j∈J_{+}} ∥T e_{j}∥, we can suppose that J = J_{+}. Fix a finite Δ ⊂ J and consider the vector subspace M_{Δ} generated by {e_{j} : j ∈ Δ}.

Since (\frac{T e_{j}}{∥T e_{j}∥})_{j∈J} is a Hilbertian family in Y with the constant c, we have

∥T h∥ ≤ c sup_{j∉Δ} ∥T e_{j}∥, ∀h ∈ S_{H}, h orthogonal to M_{Δ}.  (4.9)

Take now any finite-dimensional M ⊃ M_{Δ}. We have M = M_{Δ} + M_{2}, where M_{2} is the orthogonal complement of M_{Δ} in M; then applying (3.15) and (4.9) to T |_{M}, we get

δ(T |_{M}) ≤ \sqrt{\frac{dim M_{Δ}}{dim M}} ∥T ∥ + c sup_{j∉Δ} ∥T e_{j}∥.  (4.10)

Fix ε > 0, take a finite-dimensional M_{ε} ⊃ M_{Δ} in such a way that

\sqrt{\frac{dim M_{Δ}}{dim M_{ε}}} ∥T ∥ < ε

From (4.10), according to the choice of M_{ε}, we obtain:

δ(T |_{M}) ≤ \sqrt{\frac{dim M_{Δ}}{dim M_{ε}}} ∥T ∥ + c sup_{j∉Δ} ∥T e_{j}∥

< ε + c sup_{j∉Δ} ∥T e_{j}∥ ∀M ∈ FD(H), M ⊃ M_{ε}.

Consequently, β_{M_{ε}}(T) ≤ ε + c sup_{j∉Δ} ∥T e_{j}∥. Therefore

\overline{\overline{δ}}(T) ≤ β_{M_{ε}}(T) ≤ ε + c sup_{j∉Δ} ∥T e_{j}∥.

So \overline{\overline{δ}}(T) ≤ ε + c sup_{j∉Δ} ∥T e_{j}∥. Since the latter inequality holds for all finite subsets Δ ⊂ J, it implies

\overline{\overline{δ}}(T) ≤ ε + c inf_{Δ⊂J, card Δ<∞} sup_{j∉Δ} ∥T e_{j}∥ = ε + c u-lim sup_{j∈J} ∥T e_{j}∥.

Since ε is arbitrary, we obtain the required inequality.

The proof of (b) is similar.

Proposition 4.7. Let Y be a Hilbert space and T ∈ L(H, Y) be an orthodiagonalizable operator with diagonal (λ_{j})_{j∈J}. Then:

(a) \overline{\overline{δ}}(T) = u-lim sup_{j∈J} |λ_{j}|.

(b) \underline{\underline{δ}}(T) = u-lim inf_{j∈J} |λ_{j}|.

(c) The operator T is AMD-regular if and only if u-lim_{j∈J} |λ_{j}| exists and, in case

of AMD-regularity, the equality

\overline{δ}(T) = u-lim_{j∈J} |λ_{j}|

holds.

Proof. (a) T is ortho-diagonalizable with diagonal (λ_{j})_{j∈J} means that there are an orthonormal basis (e_{j})_{j∈J} in H and an orthonormal basis (y_{j})_{j∈J} in Y such that T e_{j} = λ_{j}y_{j}, ∀j ∈ J. The inequality \overline{\overline{δ}}(T) ≥ lim sup_{j} |λ_{j}| follows from Proposition 4.4(a) and Lemma 2.1, since Y is of type 2 with constant one. The inequality \overline{\overline{δ}}(T) ≤ lim sup_{j} |λ_{j}| follows from Proposition 4.6(a) since any orthonormal family in a Hilbert space is Hilbertian with constant one.

The proof of (b) is similar, (c) follows from (a) and (b).

Remark 4.8. (1) Evidently, if T : H → Y is a linear isometry, then \overline{δ}(T) = 1. From Proposition 4.7 we get that the converse is not true. Indeed, let H be separable and λ_{n} = 1 − \frac{1}{n+1}, n = 1, 2, …, then we have ∥T ∥ = 1, \overline{δ}(T) = 1, but T is not an isometry, even more, if x ∈ H and x ≠ 0, then ∥T (x)∥ < ∥x∥.

(2) Proposition 4.6(a,b) can give some information also in the case where Y is not necessarily a Hilbert space, e.g., if Y = l_{p}, 2 < p < ∞ and T : l_{2} → l_{p} is a diagonal operator with the diagonal (λ_{n}) with respect to natural bases, and since the natural basis of l_{p} is Hilbertian, we get \overline{\overline{δ}}(T) ≤ lim sup_{j} |λ_{j}|. This estimate is slightly better than the trivial one: \overline{\overline{δ}}(T) ≤ ∥T ∥.

4.3. AMD-numbers and compactness-like properties of operators. In this subsection we show that the AMD-numbers can be used as tools for the study of compactness-like properties of the operators acting from a Hilbert space. We begin with a proposition.

Proposition 4.9. Let Y be a Banach space.

(a) Always K(H, Y) ⊂ \{T ∈ L(H, Y) : \overline{\overline{δ}}(T) = 0\}.^{14}

14 This follows also from more delicate Theorem 4.11(b).

(b) If Y is of cotype 2, then K(H, Y) = \{T ∈ L(H, Y) : \overline{\overline{δ}}(T) = 0\} (the inclusion \{T ∈ L(H, Y) : \overline{\overline{δ}}(T) = 0\} ⊂ K(H, Y) is not true in general, see Remark 4.13).

Proof. (a) Let T ∈ K(H, Y). It is clear that if T is a finite-rank operator, then \overline{\overline{δ}}(T) = 0. If T is arbitrary, then (by Corollary 2.7) T is the norm limit of a sequence of finite-rank operators. This and the continuity of \overline{\overline{δ}} imply \overline{\overline{δ}}(T) = 0.

(b) Let T ∈ L(H, Y) and \overline{\overline{δ}}(T) = 0. Since Y is of cotype 2, by Proposition 4.4(a), we have that lim_{n} ∥T e_{n}∥ = 0 for any infinite orthonormal sequence (e_{n}) in H. Hence, by Corollary 2.7, T ∈ K(H, Y).

The next quantitative version of this proposition will be used in the last section.

Theorem 4.10. Let T ∈ L(H, Y). The following statements are valid:

(a) \overline{\overline{δ}}(T) ≤ d(T, K(H, Y)).

(b) If Y is of cotype 2, then

\overline{\overline{δ}}(T) ≥ \frac{1}{c}d(T, K(H, Y)),

where c is the cotype 2 constant of Y.

(c) If Y is a Hilbert space, then \overline{\overline{δ}}(T) = d(T, K(H, Y)).

Proof. (a) Fix any T_{0} in K(H, Y). By Proposition 4.9(a) \overline{\overline{δ}}(T_{0}) = 0. Hence

\overline{\overline{δ}}(T) = \overline{\overline{δ}}(T − T_{0} + T_{0}) ≤ \overline{\overline{δ}}(T − T_{0}) + \overline{\overline{δ}}(T_{0}) = \overline{\overline{δ}}(T − T_{0}) ≤ ∥T − T_{0}∥.

Since T_{0} is arbitrary, this inequality implies (a).

(b) According to Proposition 4.4(a) and Corollary 2.8(c) we have

c\overline{\overline{δ}}(T) ≥ sup_{(e_{n})∈O(H)} lim sup_{n} ∥T e_{n}∥ ≥ d(T, K(H, Y)).

(c) follows from (a) and (b).

The next theorem implies, in particular, that the condition \overline{\overline{δ}}(T) = 0 is satisfied if and only if T is a superstrictly singular operator.

Theorem 4.11. Let Y be a Banach space and T ∈ L(H, Y).

(a) We have:

lim_{n} b_{n}(T) ≤ \overline{\overline{δ}}(T) ≤ 3\sqrt{π} lim_{n} b_{n}(T).

(b) \overline{\overline{δ}}(T) = 0 if and only if T is superstrictly singular.

(c) If the restriction of T on some infinite-dimensional closed vector subspace of H is strictly singular, then \underline{\underline{δ}}(T) = 0.

Proof. (a) Let us show first that \overline{\overline{δ}}(T) ≥ lim_{n} b_{n}(T). Assume that lim_{n} b_{n}(T) > r > 0. Fix a natural n. Then there is E_{n} ∈ FD(X), dim E_{n} = n such that m(T |_{E_{n}}) > r. Take now any non-zero finite-dimensional M_{0} ⊂ H. An application of inequality (3.14) to M := M_{0} + E_{n}, M_{1} = M_{0} and M_{2} = E_{n} gives

β_{M_{0}}(T) ≥ δ(T |_{M_{0}+E_{n}}) ≥ \frac{δ(T |_{E_{n}})}{\sqrt{\frac{dim M_{0}}{dim E_{n}} + 1}} ≥ \frac{m(T |_{E_{n}})}{\sqrt{\frac{dim M_{0}}{n} + 1}} > \frac{r}{\sqrt{\frac{dim M_{0}}{n} + 1}}.

Since n is arbitrary, this inequality implies \overline{\overline{δ}}(T) ≥ r. As r is arbitrary, we get the needed inequality.

Let us show now \overline{\overline{δ}}(T) ≤ 3\sqrt{π} lim_{n} b_{n}(T). For this we use Proposition 3.9. We can suppose that ∥T ∥ = 1. Fix ε ∈]0, 1/2[ and a sequence (E_{n}) of finitedimensional vector subspaces of H such that lim_{n} dim(E_{n}) = ∞. Let also C be the universal constant from Proposition 3.9. Fix then a natural n. Then by means of Proposition 3.9, applied to E = E_{n}, T |_{E_{n}} and τ = 1, we can find and fix a vector subspace F_{n} ⊂ E_{n} with

dim(F_{n}) ≥ C \frac{ε^{2} dim(E_{n})}{| log ε|}  (4.11)

such that

m(T |_{F_{n}}) ≥ \frac{1}{3\sqrt{π}} δ(T |_{E_{n}}) − ε.  (4.12)

In this way we obtain a sequence (F_{n}) of finite-dimensional subspaces of H, which depends on T and on ε. Put k_{n} := dim(F_{n}), n = 1, 2, … Clearly, (4.12) implies

b_{k_{n}}(T) ≥ \frac{1}{3\sqrt{π}} δ(T |_{E_{n}}) − ε, n = 1, 2, ….  (4.13)

Since lim_{n} dim(E_{n}) = ∞, from (4.11) we conclude that lim k_{n} = ∞. This and (4.13) imply

lim_{n} b_{n}(T) = lim_{n} b_{k_{n}}(T) ≥ \frac{1}{3\sqrt{π}} lim sup_{n} δ(T |_{E_{n}}) − ε, n = 1, 2, …. (4.14)

From (4.14), since (E_{n}) is arbitrary, according to Proposition 4.2, we obtain lim_{n} b_{n}(T) ≥ \frac{1}{3\sqrt{π}} \overline{\overline{δ}}(T)−ε. Since ε is also arbitrary, we get \overline{\overline{δ}}(T) ≤ 3\sqrt{π} lim_{n} b_{n}(T).

(b) follows from (a).

(c) Fix a finite-dimensional vector subspace M_{0} ⊂ H and a number ε > 0. Let X ⊂ H be a closed vector subspace such that T |_{X} : X → Y is strictly

singular. Then, by [52, (1.9.1)], we can find and fix an infinite-dimensional closed vector subspace H_{0} ⊂ X such that ∥T |_{H_{0}}∥ < ε. Take now an infinite sequence (M_{n}) of finite-dimensional subspaces of H_{0} such that dim M_{n} = n, n = 1, 2, . . .. According to (3.15), for any fixed n we can write

δ(T |_{M_{0}+M_{n}}) ≤ \sqrt{\frac{dim M_{0}}{dim M_{0} + dim M_{n}}} ∥T |_{M_{0}}∥ + δ(T |_{M_{0}})

≤ \sqrt{\frac{dim M_{0}}{dim M_{0} + n}} ∥T ∥ + ∥T |_{M_{n}}∥ < \sqrt{\frac{dim M_{0}}{dim M_{0} + n}} ∥T ∥ + ε.

From this we get

α_{M_{0}}(T) ≤ δ(T |_{M_{0}+M_{n}}) < \sqrt{\frac{dim M_{0}}{dim M_{0} + n}}∥T ∥ + ε, n = 1, 2, ….

Hence α_{M_{0}}(T) ≤ ε and, since M_{0} is arbitrary, \underline{\underline{δ}}(T) ≤ ε. As ε is also arbitrary, we get \underline{\underline{δ}}(T) = 0.

The next “AMD-free” statement of independent interest is not a consequence of Theorem 2.18 because it does not require the existence of an unconditional basis.

Theorem 4.12. Let H be a Hilbert space and Y be any Banach space of cotype 2. Then SSS(H, Y) = K(H, Y).

Proof. This follows from Theorem 4.11(b) and Proposition 4.9.

Remark 4.13. (1) Let 2 < p < ∞ and let T_{2,p} : ℓ_{2} → ℓ_{p} be the natural embedding. Then by Proposition 2.10(d) we have that T_{2,p} is a SSS-operator and therefore, by Theorem 4.11, \overline{\overline{δ}}(T_{2,p}) = 0^{15}. Since T_{2,p} is not compact, it turns out that Proposition 4.9, as well as Theorem 4.12, may not be true when Y is not of cotype 2. However, in general, for a Banach space Y the validity even of the inclusion SS(ℓ_{2}, Y) ⊂ K(ℓ_{2}, Y) may not imply that Y is of cotype 2 (see Remark 2.19(5)).

(2) Let us say for a moment that a Banach space Y is a SSSC-space if SSS(ℓ_{2}, Y) ⊂ K(ℓ_{2}, Y). By Theorem 4.12 any cotype 2 Banach space is a SSSCspace. Evidently, any Banach space with the Schur property is a SSSC-space too. It would be interesting to find an internal characterization of SSSC-spaces.

5. AMD-Numbers and Bounds of the Essential Spectrum

5.1. Spectrum and spectral radius. Let 𝔄 be a Banach algebra over 𝕂 with unit e. The spectrum σ(a) of an element a ∈ 𝔄 is defined as the set of all λ ∈ 𝕂 such that the element a − λe is not invertible in 𝔄. For an arbitrary element a, σ(a) is a compact subset of 𝕂 which is non-empty when 𝕂 = ℂ (by Gelfand’s theorem), but which may be empty when 𝕂 = ℝ.

15 The equality \overline{\overline{δ}}(T_{2,p}) = 0 can also be proved directly, namely it can be shown that for any vector subspace E ⊂ ℓ_{2} with dim E = n one has δ(T_{2,p}|_{E}) ≤ a_{p}(γ_{𝕂})(dim E)^{1/p−1/2}.

For an element a ∈ 𝔄 with non-empty spectrum, its spectral radius r(a) is defined by the equality

r(a) = sup{|λ| : λ ∈ σ(a)}.

Notice that r(a) ≤ ∥a∥ for any a ∈ 𝔄. Clearly, if for a given a ∈ 𝔄 we have σ(a) ≠ ∅ and σ(a) ⊂ ℝ, then σ(a) ⊂ [min(σ(a)), max(σ(a))].

The following assertions are easy to prove:

(Spec1) If a ∈ 𝔄 is such that σ(a) ≠ ∅ and λ ∈ 𝕂, then σ(a − λe) =

σ(a) − λ(:= {λ′ − λ : λ′ ∈ σ(a)}).

(Spec2) If a ∈ 𝔄 is such that σ(a) ≠ ∅ and a is invertible, then 0 ∉ σ(a)

and σ(a^{−1}) = (σ(a))^{−1}(:= {λ^{−1} : λ ∈ σ(a)}). (Spec3) If a ∈ 𝔄 is such that σ(a) ≠ ∅ and λ ∉ σ(a), then σ((a−λe)^{−1}) ≠ ∅

and hence r((a − λe)^{−1}) is defined.

Taking into account (Spec3) let us say that an element a ∈ 𝔄 is normal-like if σ(a) ≠ ∅ and r((a − λe)^{−1}) = ∥(a − λe)^{−1}∥ ∀λ ∈ 𝕂 \ σ(a).

Proposition 5.1. Let 𝔄 be a real or complex Banach algebra with unit, (a_{n}) be a sequence of elements of 𝔄 such that σ(a_{n}) ≠ ∅ ∀ n, a ∈ 𝔄 and a_{n} → a in 𝔄. Then:

(a) If (λ_{n})_{n∈ℕ} is a sequence of scalars such that λ_{n} ∈ σ(a_{n}) for all n, then (λ_{n}) is a bounded sequence and its cluster points belong to σ(a). In particular, σ(a) ≠ ∅.

(b) lim sup_{n} r(a_{n}) ≤ r(a). Moreover, if σ(a) ⊂ ℝ and σ(a_{n}) ⊂ ℝ for all n ∈ ℕ, then

lim sup_{n}(max(σ(a_{n}))) ≤ max(σ(a)),

and

lim inf_{n}(min(σ(a_{n}))) ≥ min(σ(a)).

(c) If the elements a_{n}, n = 1, 2, … are normal-like, then for any λ ∈ σ(a) we have lim_{n} d(λ, σ(a_{n})) = 0.

(d) If the assumptions of (c) are satisfied then r(a) = lim_{n} r(a_{n}). Moreover, if σ(a) ⊂ ℝ and σ(a_{n}) ⊂ ℝ for all n ∈ ℕ, then

lim_{n} max(σ(a_{n})) = max(σ(a))

and

lim_{n} min σ(a_{n}) = min σ(a).

Proof. (a) Evidently, |λ_{n}| ≤ ∥a_{n}∥ ≤ sup_{n} ∥a_{n}∥ < ∞ for all n ∈ ℕ. Hence (λ_{n}) is a bounded sequence. Now it is easy to prove that any cluster point of (λ_{n}) belongs to σ(a).

(b) follows easily from (a).

(c) Suppose λ ∈ σ(a), but lim_{n} d(λ, σ(a_{n})) ≠ 0. Then there exists a strictly positive number ε such that d(λ, σ(a_{n})) ≥ ε for infinitely many values of n. There is no loss in simplifying the notation and assuming that

d(λ, σ(a_{n})) ≥ ε, n = 1, 2, ….

Hence

λ ∉ σ(a_{n})), n = 1, 2, ….

From this, via (Spec2) and according to the definition of the spectral radius, we get

r((a_{n} − λe)^{−1}) ≤ ε^{−1}, n = 1, 2, ….

The latter relation, together with the supposition that the considered elements are normal-like, gives

∥(a_{n} − λe)^{−1}∥ ≤ ε^{−1}, n = 1, 2, ….  (5.1)

Put b_{n} = a_{n} − λe, n = 1, 2, . . .. Using the relation

b_{n}^{−1} − b_{m}^{−1} = b_{m}^{−1}(a_{m} − a_{n})b_{n}^{−1}, , n, m = 1, 2, …,

and (5.1), we obtain

∥b_{n}^{−1} − b_{m}^{−1}∥ ≤ ∥b_{m}^{−1}∥ · ∥a_{m} − a_{n}∥ · ∥b_{n}^{−1}∥ ≤ ε^{−2}∥a_{m} − a_{n}∥, n, m = 1, 2, …,

and since a_{n} → a, it follows that the squence (b_{n}^{−1}) converges to some element b. Then

(a − λe)b = lim_{n}(a_{n} − λe) lim_{n} b_{n}^{−1} = lim_{n}(a_{n} − λe)(a_{n} − λe)^{−1} = e

and, similarly, b(a − λe) = e. This a − λe is invertible and λ ∉ σ(a). A contradiction.

(d) follows in a standard way from (c) and (b).

Remark 5.2. The reader familiar with the notion of convergence of sets in Kuratowski’s sense (see [32, p. 335–340]) can observe easily that Proposition 5.1(a) is a direct formulation of the well-known fact: the set-valued function σ is upper semicontinuous on 𝔄. Similarly, Proposition 5.1(c) together with Proposition 5.1(a) implies that this function is continuous on the (closed) set 𝔄_{0} consisting of normal-like elements of 𝔄. Our proof is taken from [25] (see the solution of problem 105) where the statement is proved for normal operators in a complex Hilbert space. We see that the proof works for normal-like elements in real algebra too.

5.2. Essential spectrum. Let H be an infinite-dimensional real or complex Hilbert space, then L(H) is a Banach algebra with unit. It follows that when H is a complex Hilbert space, any T ∈ L(H) has a non-empty spectrum. It is well-known that when T is any self-adjoint operator in a real or complex Hilbert space, the fact that σ(T) is non-empty can be proved easily without any reference to Gelfand’s theorem. Also the following facts have direct and easy proofs.

Let H be a Hilbert space over 𝕂, T ∈ L(H) and T^{∗} = T. Then:

• (T x|x) ∈ ℝ ∀x ∈ H and σ(T) ⊂ ℝ.

• σ(|T |) = {|λ| : λ ∈ σ(T)} and r(T) = ∥T ∥.

• sup{(T x|x) : x ∈ S_{H}} = max σ(T) and inf{(T x|x) : x ∈ S_{H}} =

min σ(T).

The set K(H) is a closed two-sided ideal of L(H). Let π:L(H)→L(H)/K(H) be the canonical map. Then it is known that L(H)/K(H) too is a Banach algebra with unit π(I) and with respect to the quotient norm

∥π(T)∥ = d(T, K(H)), T ∈ L(H),

which is called the Calkin algebra. For a given T ∈ L(H) the essential spectrum σ_{e}(T) is by definition the spectrum σ(π(T)) of its canonical image π(T) in the Calkin algebra ([16, p. 358]).

Notice that for any T ∈ L(H) we have σ_{e}(T) ⊂ σ(T). If H is a complex Hilbert space, then for any T ∈ L(H) the essential spectrum is non-empty (this is so because in any complex Banach algebra with unit any element has a non-empty spectrum). In what follows, we need the fact that any self-adjoint operator in a real Hilbert space also has a non-empty essential spectrum. We shall derive this from the corresponding fact for strictly diagonalizable operators via the following lemma which asserts that such operators are sufficiently many.

Lemma 5.3. Let H be a real or complex infinite-dimensional Hilbert space. The following assertions hold:

(a) Let S ∈ L(H) be any self-adjoint operator. Then there is a sequence D_{n} : H → H, n ∈ ℕ of self-adjoint strictly ortho-diagonalizable operators such

that lim_{n} ∥D_{n} − S∥ = 0.

(b) If Y is another Hilbert space, then the set of all ortho-diagonalizable operators is norm dense in L(H, Y).

Proof. (a) Fix S. According to the spectral representation theorem which holds in the case of real Hilbert space too [63, Theorem 7.17, p.191], there exists a resolution of identity E on Borel subsets of Λ := σ(S) such that

S = ∫_{Λ} λE(dλ).

Notice that this integral can be computed as follows: fix a sequence f_{n} : Λ → Λ, n ∈ ℕ, of simple measurable functions such that sup{|f_{n}(λ) − λ|, λ ∈ Λ} → 0 when n → ∞; put

D_{n} = ∫_{Λ} f_{n}(λ) dE(λ), n ∈ ℕ,  (5.2)

then D_{n} : H → H, n ∈ ℕ are self-adjoint linear operators and lim_{n} ∥S − D_{n}∥=0.

The properties of the resolution of identity easily imply that each of the operators D_{n}, n ∈ ℕ, is strictly ortho-diagonalizable. Therefore (D_{n}) is the required sequence for S.

(b) Fix T ∈ L(H, Y). Let H_{1} be the closure of |T |(H), where |T | = (T^{∗}T)^{\frac{1}{2}} . There is an isometry U : H_{1} → Y such that U|T |h = T h ∀ h ∈ H. The restriction S of |T | on H_{1} is a self-adjoint positive linear operator acting in the Hilbert space H_{1}. Since H = ker |T | + H_{1}, we have T = USP_{1}, where P_{1} : H → H_{1} is an orthogonal projection. Let now D_{n}, n ∈ ℕ, be the orthodiagonalizable operators constructed for S according to (a). Now it is easy to check that the operators UD_{n}P_{1}, n ∈ ℕ, are ortho-diagonalizable and lim_{n} ∥T − UD_{n}P_{1}∥ = 0.^{16}

Lemma 5.4. Let H be a real or complex Hilbert space and let T ∈ L(H) be a strictly ortho-diagonalizable operator with the diagonal (λ_{j})_{j∈J}. The following assertions are valid:

(a) 0∉σ_{e}(T) if and only if u-lim inf_{j∈J} |λ_{j}| > 0. Moreover, if 0∉σ_{e}(T), then (π(T))^{−1} = π(B), where B ∈ L(H) is a strictly diagonalizable operator with respect to the same orthonormal basis as T.

(b) σ_{e}(T) coincides with the set of numbers λ which have the form λ = lim λ_{j_{n}} where (j_{n}) is a sequence of distinct elements of J such that lim_{n} λ_{j_{n}} exists. Equivalently, if λ is a number, then λ ∈ σ_{e}(T) if and only if u- lim inf_{j∈J} |λ_{j} − λ| = 0. In particular, σ_{e}(T) ≠ ∅.

(c) σ_{e}(|T |) = {|λ| : λ ∈ σ_{e}(T)} and r(π(T)) = ∥π(T)∥.

(d) If 0 ∉ σ_{e}(T) then r((π(T))^{−1}) = ∥(π(T))^{−1}∥.

(d′) The element π(T) is normal-like in the Calkin algebra L(H)/K(H).

(e) If A ∈ L(H) is any self-adjoint operator, then σ_{e}(A) ≠ ∅.

Proof. (a) We can suppose that T is a diagonal operator in an orthonormal basis (e_{j})_{j∈J} of H. Then the first part of (a) can be proved as a similar assertion in [16] (see the proof of the equivalence (a)⇔(e) of Theorem 2.3 on pp. 350-351); we note only that in our case of a diagonal operator the proof is easier and it works also in the case of a real Hilbert space. Let now 0 ∉ σ_{e}(T), then according to the first part we have lim inf_{j∈J} |λ_{j}| > 0. This implies that inf_{j∉Δ} |λ_{j}| > 0 for some finite subset Δ ⊂ J. Define now B ∈ L(H) in the following way: B(e_{j}) = 0 if j ∈ Δ and B(e_{j}) = \frac{1}{λ_{j}} e_{j} if j ∉ Δ. Then B is a diagonal operator and it is easy to see that π(B) is the inverse of π(T) in the Calkin algebra.

16 We are indebted to Professor S. Kwapień who communicated to us the idea of deriving this lemma from the spectral theorem. We do not know whether there exists a proof which avoids spectral integrals.

(b) The first part of (b) follows directly from the first part of (a). From the first part of (b) it is clear also that the essential spectrum of T is not empty.

(c) Since |T | is a strictly diagonalizable operator with diagonal (|λ_{j}|), according to (b) we have the first equality from (c). Now again from (b) and from the definition of lim sup_{j} |λ_{j}| it follows that r(π(T)) = sup σ_{e}(|T |) = lim sup_{j} |λ_{j}|. This and equality (2.6) of Corollary 2.21 imply r(π(T)) = ∥π(T)∥.

(d) The inequality r((π(T))^{−1}) ≤ ∥(π(T))^{−1}∥ is valid for general reasons. The inverse inequality will be proved if we can find λ in σ((π(T))^{−1}) such that |λ| = ∥(π(T))^{−1}∥. Take B from (a). If we apply now the second equality of (c) to the operator B, we can find λ ∈ σ(π(B)) = σ((π(T))^{−1}) such that |λ| = ∥π(B)∥ = ∥(π(T))^{−1}∥ and (d) is proved.

(d′) Let λ ∈ 𝕂 \ σ_{e}(T). Put T_{λ} = T − λ I. Then T_{λ} is strictly ortho-diago-nalizable and 0 ∉ σ_{e}(T_{λ}). Then, by (d), applied to T_{λ}, we get: r((π(T_{λ}))^{−1}) = ∥(π(T_{λ}))^{−1}∥. Hence π(T) is normal-like.

(e) Via Lemma 5.3(a) we can find a sequence (T_{n}) of strictly ortho-diagona-lizable operators in H such that lim_{n} ∥T_{n} − A∥ = 0. It follows that lim_{n} ∥π(T_{n}) − π(A)∥ = 0. Now an application of (b) and Proposition 5.1(a) to the Calkin algebra 𝔄 = L(H)/K(H) gives that σ_{e}(A) := σ(π(A)) ≠ ∅.

5.3. Bounds of the essential spectrum. We begin with an assertion which shows that the mean dilatation numbers in the case of Hilbert spaces depend only on the modulus of operators.

Lemma 5.5. Let Y be a Hilbert space and T ∈ L(H, Y). Then \overline{\overline{δ}}(T) = \overline{\overline{δ}}(|T |), \underline{\underline{δ}}(T) = \underline{\underline{δ}}(|T |). Consequently, T is AMD-regular if and only if |T | is such and, in the case of AMD-regularity, we have \overline{δ}(T) = \overline{δ}(|T |).

Proof. This follows easily from the definitions and from the observation that the equality ∥T x∥ = ∥|T |(x)∥ holds for all x ∈ H.

The next assertion, which is the main result of this section, shows that by using mean dilatation numbers it is possible to describe the bounds of the essential spectrum.

Theorem 5.6. Let H, Y be infinite-dimensional Hilbert spaces, T ∈ L(H, Y) be an arbitrary operator. Then

\overline{\overline{δ}}(T) = max σ_{e}(|T |), \underline{\underline{δ}}(T) = min σ_{e}(|T |).

Consequently, T is AMD-regular if and only if σ_{e}(|T |) consists of one point λ_{0}, and, in such a case, \overline{δ}(T) = λ_{0}.

Proof. Suppose first that T is an ortho-diagonalizable operator with diagonal (λ_{j}). Then |T | is a strictly ortho-diagonalizable operator with diagonal (|λ_{j}|). Hence by Lemma 5.4, Lemma 2.1 and Corollary 2.21 (through equality (2.6)) we can write the equalities:

max σ_{e}(|T |) = u-lim sup_{j} |λ_{j}| = \overline{\overline{δ}}(T), min σ_{e}(|T |) = u-lim inf_{j} |λ_{j}| = \underline{\underline{δ}}(T).

Therefore the assertion is true in this case.

Let now T be an arbitrary operator. By Lemma 5.3(b), there is a sequence (T_{n}) of ortho-diagonalizable operators, T_{n} ∈ L(H, Y), such that lim_{n} ∥T − T_{n}∥ = 0. It follows that lim_{n} ∥|T | − |T_{n}|∥ = 0. This implies lim_{n} ∥π(|T |) − π(|T_{n}|)∥ = 0. Notice now that according to Lemma 5.4(d′) the elements π(|T_{n}|), n = 1, 2, …, are normal-like in the Calkin algebra L(H)/K(H). So we can apply Proposition 5.1(d) for the elements π(|T |), π(|T_{n}|), n ∈ ℕ, and write max σ_{e}(|T_{n}|) → max σ_{e}(|T |) and min σ_{e}(|T_{n}|) → min σ_{e}(|T |) (let us emphasize that this is the crucial point of the proof, only here the delicate Proposition 5.1(d) is used). Now, using this, Lemma 5.5 and the continuity of functionals \overline{\overline{δ}} and \underline{\underline{δ}} (see Proposition 4.1), we obtain:

\overline{\overline{δ}}(T) = \overline{\overline{δ}}(|T |) = lim_{n} \overline{\overline{δ}}(|T_{n}|) =
= lim_{n} max σ_{e}(|T_{n}|) = max σ_{e}(|T |).

Similarly,

\underline{\underline{δ}}(T) = \underline{\underline{δ}}(|T |) = lim_{n} \underline{\underline{δ}}(|T_{n}|) =
= lim_{n} min σ_{e}(|T_{n}|) = min σ_{e}(|T |).

The assertion about \overline{δ} is now immediate.

Remark 5.7. The exact bounds of the essential spectrum for an operator in a complex Hilbert space had been studied earlier in several papers, see, e.g., [6], [41], [65]. For the case of a complex Hilbert space H and the description of the essential spectrum by means of Weyl’s criterion [57, p.237, Theorem VII.12] together with the observation that for any T ∈ L(H) one has ∥π(T)∥ ∈ σ_{e}(|T |) (which follows from the general result concerning the spectrum of an element in a C^{∗}-algebra), we get

\overline{\overline{δ}}(T) = max σ_{e}(|T |).

As the reader can see, our proof avoids the usage of the rather delicate fact that the Calkin algebra is a C^{∗}-algebra.

Acknowledgments

(1) The authors express special thanks to Professor Anatolij Plichko for his attention and interest in this paper during his visit University of Vigo (December, 2000) and later; the final versions of Theorems 2.18(a) and 4.11(b) are partially obtained in collaboration with him.

(2) The third author was supported by Georgian Academy of Sciences grant No. 1.16 and by “Axudas para o financiamento de estadias de profesores visitantes Xunta 2000(D.O.G. do 25.02.2000)” during his stay at Departamento de Matemática Aplicada Universidade de Vigo (Galicia, Spain) in the academic year 2000-2001. He is grateful to this University for the hospitality and good conditions for work.

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(Received 18.07.2001; revised 12.12.2001)

Authors’ addresses:

A. Castejón and E. Corbacho Departamento de Matemática Aplicada Universidad de Vigo 36200 Vigo, Galicia España E-mail: acaste@dma.uvigo.es

corbacho@uvigo.es

V. Tarieladze Muskhelishvili Institute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 380093 Georgia E-mail: tar@gw.acnet.ge