Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 3–11

DUAL RESULTS OF FACTORIZATION FOR OPERATORS

Manuel Gonzalez

Universidad de Cantabria, Departamento de Matematicas Facultad de Ciencias, E-39071 Santander, Spain; mgonzalez@ccucvx.unican.es

Abstract. We study the duality properties of the well-known DFJP factorization of operators [3] by means of a refinement of it. Given an operator T: X → Y we consider a decomposition T = jUk, where U: E → F is an isomorphism, and j, Uk are the factors in the DFJP factorization.

If T^{∗} is the conjugate operator of T , and T̄: X^{∗∗}/X → Y^{∗∗}/Y is the operator given by T̄(x + X) := T^{∗∗}x + Y (x ∈ X^{∗∗}), then we show that the decompositions of T^{∗} and T̄ are precisely k^{∗}U^{∗}j^{∗} = (jUk)^{∗} and j̄Ūk̄. From this result we derive several consequences. For example, we detect new operator ideals with the factorization property, we characterize operators whose conjugate is Rosenthal, and using a result of Valdivia [11] we show that an operator T such that T̄ has separable range can be decomposed as T = S + K, where S^{∗∗} has separable range and K is weakly compact.

0. Introduction

For a (continuous linear) operator T ∈ L(X, Y) we shall introduce a decomposition T = jUk in which U is an isomorphism, j is an injective tauberian operator, and k is a cotauberian operator with dense range.

This decomposition is inspired by the equivalent versions of the real interpolation method of Banach spaces [2], and it is a refinement of the well-known DFJP factorization introduced in [3] which factorizes T in two factors: j and Uk. Moreover, the factorization of T in two factors jU and k was considered in [5].

We show that k^{∗}U^{∗}j^{∗} coincides with the decomposition of the conjugate operator T^{∗} ∈ L(Y^{∗}, X^{∗}), and j̄Ūk̄ coincides with the decomposition of the operator T̄ ∈ L(X^{∗∗}/X, Y^{∗∗}/Y). Moreover, if T belongs to a closed operator ideal 𝒜 , then k and j belong to the injective hull and the surjective hull of 𝒜 , respectively. In this way the decomposition of an operator makes clear the duality properties and the symmetry of the DFJP factorization.

As an application we obtain necessary conditions for the factorization property for an operator ideal, and we show that some operator ideals defined in terms of T^{∗} or T̄ verify this property. Also we characterize the class of operators whose conjugate is Rosenthal as those operators which factor into a Banach space containing no copies of l_{1} and no quotients isomorphic to c_{0} , and using a result of [11] we show that an operator T such that T̄ has separable range can be decomposed as T = S + K, where S^{∗∗} has separable range and K is weakly compact.

1991 Mathematics Subject Classification: Primary 47D50; Secondary 46B70.

Supported in part by DGICYT Grant PB88-0417.

Notations. X, Y , Z will be Banach spaces, and B_{X} the closed unit ball of X. For an operator T ∈ L(X, Y) we shall denote T^{∗} ∈ L(Y^{∗}, X^{∗}) the conjugate operator, and T̄ ∈ L(X^{∗∗}/X, Y^{∗∗}/Y) the operator defined by

T̄(z + X) := T^{∗∗}z + Y (z ∈ X^{∗∗}).

The properties of T̄ have been studied in [12].

An operator ideal 𝒜 is said to be closed if 𝒜 (X, Y) is closed in L(X, Y) for every pair of spaces X, Y .

𝒜 is injective if given T ∈ L(X, Y) and an injection (isomorphism into) J ∈ L(Y, Z) we have JT ∈ 𝒜 implies T ∈ 𝒜 .

𝒜 is surjective if given T ∈ L(X, Y) and a surjection (surjective operator) Q ∈ L(Z, X) we have T Q ∈ 𝒜 implies T ∈ 𝒜 .

𝒜 has the factorization property if every T ∈ 𝒜 factors through a Banach space Z such that the identity I_{Z} belongs to 𝒜 .

For an account of the theory of operator ideals we refer to [9].

1. Construction of the decomposition

In this section, given Banach spaces X and Y , and an operator T ∈ L(X, Y) we shall construct the decomposition T = jUk.

For each positive integer n we denote

p_{n}(x) := 2^{n} ∥T x∥ + 2^{−n} ∥x∥ (x ∈ X),
q_{n} the gauge of the set 2^{n}T B_{X} + 2^{−n}B_{Y} .

Clearly p_{n} and q_{n} are norms in X and Y respectively, equivalent to the initial ones.

We shall consider also the Banach spaces

l_{2}(X, p_{n}) := {(x_{n})/x_{n} ∈ X, ( ∑_{n=1}^{∞} p_{n}(x_{n})^{2})^{1/2} < ∞} and l_{2}(Y, q_{n})

endowed with its natural norms.

1.1. Lemma. For every (x_{k}) ∈ l_{2}(X, p_{n}), the series ∑_{n=1}^{∞} T x_{n} is absolutely convergent in Y . Moreover

N := {(x_{n}) ∈ l_{2}(X, p_{n})/ ∑_{n=1}^{∞} T x_{n} = 0} is a closed subspace of l_{2}(X, p_{n}).

Proof. Note that

∥T x_{k}∥ < 2^{−k} ∥(x_{n})∥ for every k and every (x_{n}) ∈ l_{2}(X, p_{n}).

Hence the series ∑_{n=1}^{∞} T x_{n} is absolutely convergent, and N is closed.

The above lemma allows us to introduce the intermediate spaces E, F of the decomposition, and two of the factors.

E := l_{2}(X, p_{n})/N, and
k: x ∈ X → (x, 0, 0, …) + N ∈ E.
F := {(y_{k}) ∈ l_{2}(Y, q_{n})/y_{k} = y_{1} for every integer k}, and
j: (y, y, y, …) ∈ F → y ∈ Y.

Our next result will allow us to connect j and k.

1.2. Theorem. The map U: E → F defined by

U((x_{n}) + N) := ( ∑_{n=1}^{∞} T x_{n}, ∑_{n=1}^{∞} T x_{n}, …)

is an isomorphism of E onto F.

Proof. First we prove that U is well-defined and continuous. Fix (x_{n}) ∈ l_{2}(X, p_{n}). Denoting

c_{m} := ∥2^{−m} ∑_{k=1}^{m} x_{k}∥ = ∥ ∑_{k=0}^{m−1} 2^{−k}2^{−(m−k)}x_{m−k}∥

we have

( ∑_{n=1}^{∞} c_{n}^{2})^{1/2} ≤ ∑_{k=0}^{∞} 2^{−k}( ∑_{n=1}^{∞} ∥2^{−n}x_{n}∥)^{1/2} ≤ 2∥(x_{n})∥.

Also, if

d_{m} := ∥2^{m} ∑_{k=m+1}^{∞} T x_{k}∥ = ∥ ∑_{k=1}^{∞} 2^{−k}2^{m+k}T x_{m+k}∥

we have

( ∑_{n=1}^{∞} d_{n}^{2})^{1/2} ≤ ∑_{k=1}^{∞} 2^{−k}( ∑_{n=1}^{∞} ∥2^{n}T x_{n}∥^{2})^{1/2} ≤ ∥(x_{n})∥.

Note that ∑_{k=1}^{m} x_{k} ∈ 2^{m}c_{m}B_{X}; hence ∑_{k=1}^{m} T x_{k} ∈ 2^{m}c_{m}T B_{X}. Analogously, ∑_{k=m+1}^{∞} T x_{k} ∈ 2^{−m}d_{m}B_{Y} .

Then we have q_{m}( ∑_{n=1}^{∞} T x_{n}) ≤ max{c_{m}, d_{m}}; hence

( ∑_{m=1}^{∞} q_{m}( ∑_{n=1}^{∞} T x_{n})^{2})^{1/2} ≤ 2 ∥(x_{n})∥ .

We conclude that

U((x_{n}) + N) = ( ∑_{n=1}^{∞} T x_{n}, ∑_{n=1}^{∞} T x_{n}, …) ∈ l_{2}(Y, q_{n})  and  ∥U∥ ≤ 2.

It follows immediately from the definition that U is injective. It remains only to prove that U is surjective.

Given (y, y, y, …) ∈ F and denoting b_{n} := q_{n}(y), for any ε > 0 we have y ∈ (1 + ε)b_{n}(2^{n}T B_{X} + 2^{−n}B_{Y}).

Then we can write y = T u_{n} + v_{n} with ∥u_{n}∥ ≤ 2^{n}(1 + ε)b_{n} and ∥v_{n}∥ ≤ 2^{−n}(1 + ε)b_{n}.

Note that T u_{n} converges to y, since b_{n} converges to 0. We take x_{1} = u_{1} and x_{n} := u_{n} − u_{n−1} for n > 1.

Obviously ∑_{n=1}^{∞} T x_{n} converges to y in Y . Moreover, 2^{−n} ∥x_{n}∥ ≤ 2(1 + ε)b_{n} and, since T x_{n} = v_{n−1} − v_{n} for n > 1, 2^{n}∥T x_{n}∥ ≤ 3(1 + ε)b_{n}.

In this way we obtain ( ∑_{n=1}^{∞} p_{n}(x_{n})^{2})^{1/2} ≤ 4(1 + ε)( ∑_{n=1}^{∞} q_{n}(y)^{2}) < ∞; hence (x_{n}) ∈ l_{2}(X_{n}, p_{n}) and U((x_{n}) + N) = (y, y, y, …).

The proof is finished.

1.3. Definition. We shall call jUk the decomposition of T. It is immediate to check that jUkx = T x for every x ∈ X.

[figure: \begin{matrix}X & Y \\ E & F \end{matrix}; k; T; U; j]

Note that j, Uk are the factors, and F the intermediate space in the DFJP decomposition [3], and jU, k are the factors, and E the intermediate space in the factorization considered in [5].

Next we shall show the duality properties of the decomposition. Given the conjugate T^{∗} ∈ L(Y^{∗}, X^{∗}) of T ∈ L(X, Y), for every positive

integer n we shall denote the equivalent norms associated with T^{∗} in the following way.

p_{n}^{∗}(g) := 2^{n}∥T^{∗}g∥ + 2^{−n} ∥g∥ (g ∈ Y^{∗}),
q_{n}^{∗} the gauge of the set 2^{n}T^{∗}B_{Y^{∗}} + 2^{−n}B_{X^{∗}}.

Moreover, for subsets A ⊂ X and V ⊂ X^{∗}, we shall denote by A^{∘} and ^{∘}V the polar subsets given by

A^{∘} := {f ∈ X^{∗}/|f(x)| ≤ 1 for x ∈ A}  and
∘V := {x ∈ X/|f(x)| ≤ 1 for f ∈ V }.

1.4. Lemma. We have (X, p_{n})^{∗} = (X^{∗}, q_{n}^{∗}) and (Y, q_{n})^{∗} = (Y^{∗}, p_{n}^{∗}) isometrically.

Proof. Denoting Z := X × Y endowed with the norm ∥(x, y)∥ := ∥x∥ + ∥y∥, we consider the auxiliary operator S ∈ L(X, Z) defined by Sx := (2^{−n}x, 2^{n}T x), and its conjugate S^{∗} ∈ L(Z^{∗}, X^{∗}), which is given by S^{∗}(f, g) = 2^{−n}f + 2^{n}T^{∗}g.

We have that the unit ball of (X, p_{n}) coincides with S^{−1}B_{Z}. Then, as Z^{∗} is X^{∗} × Y^{∗} with the supremum norm, the unit ball of (X, p_{n})^{∗} is

(S^{−1}B_{Z})^{∘} = S^{∗}B_{Z^{∗}} = 2^{n}T^{∗}B_{Y^{∗}} + 2^{−n}B_{X^{∗}},

which coincides with the unit ball of (X^{∗}, q_{n}^{∗}).

The other part of the lemma can be proved in a similar manner.

Next we establish the main duality properties of the decomposition.

1.5. Theorem. Suppose jUk is the decomposition of T. Then k^{∗}U^{∗}j^{∗} is the decomposition of T^{∗}.

Proof. We begin showing that j^{∗} coincides with the first term of the decomposition of T^{∗}.

Considering F as a closed subspace of l_{2}(Y, q_{n}), we have that j^{∗} acts from Y^{∗} into l_{2}(Y, q_{n})^{∗}/F^{∘}.

By Lemma 1.4, we can identify l_{2}(Y, q_{n})^{∗} and l_{2}(Y^{∗}, p_{n}^{∗}). Moreover it is not difficult to check that under this identification

F^{∘} = {(g_{k}) ∈ l_{2}(X^{∗}, p_{n}^{∗})/ ∑_{n=1}^{∞} T^{∗}g_{n} = 0},

and for every g ∈ Y^{∗} and (y, y, y, …) ∈ F we have

(j^{∗}g)(y, y, y, …) = g(y) = (g, 0, 0, …)(y, y, y, …);

hence j^{∗}g = (g, 0, 0, …) + F^{∘}, as we wanted to prove.

Next we show that k^{∗} is the third term in the decomposition of T^{∗}.

The operator k^{∗} acts from E^{∗} = (l_{2}(X, p_{n})/N)^{∗} = N^{∘}, which can be identified with the subspace

{(f_{k}) ∈ l_{2}(X^{∗}, q_{n}^{∗})/f_{k} = f_{1} for every integer k},

into X^{∗}. Moreover, for every (f, f, f, …) ∈ l_{2}(X^{∗}, q_{n}^{∗}) and x ∈ X we have (k^{∗}(f, f, f, …))(x) = f(x); hence k^{∗}(f, f, f, …) = f.

Finally, U^{∗} is an isomorphism and verifies k^{∗}U^{∗}j^{∗} = T^{∗}. Hence it is the second term, since j^{∗} has dense range, and k^{∗} is injective.

1.6. Theorem. Suppose jUk is the decomposition of T. Then j̄Ūk̄ is the decomposition of T̄.

Proof. If Z_{n} is a sequence of Banach spaces, then the map

(w_{n}) + l_{2}(Z_{n}) ∈ l_{2}(Z_{n}^{∗∗})/l_{2}(Z_{n}) → (w_{n} + Z_{n}) ∈ l_{2}(Z_{n}^{∗∗}/Z_{n})

defines a bijective isometry.

Also, for a closed subspace M of X, we can identify

M^{∗∗}/M = (M^{∘∘} + X)/X,  and  (X/M)^{∗∗}/(X/M) = X^{∗∗}/(M^{∘∘} + X).

The operator j̄ acts from F^{∗∗}/F into Y^{∗∗}/Y , and we have

F^{∘∘} = {(z_{k}) ∈ l_{2}(Y^{∗∗}, q_{n})/z_{k} = z_{1} for every k}

and

F^{∗∗}/F = {(z_{k} + Y_{k}) ∈ l_{2}((Y, q_{n})^{∗∗})/l_{2}(Y, q_{n}) / z_{k} = z_{1} for every k};

hence j̄(z_{1} + Y_{k}) = j^{∗∗}(z_{1}) + Y = z_{1} + Y , and we conclude that j̄ coincides with the third term in the decomposition of T̄.

Analogously, we can identify E^{∗∗}/E with l_{2}((X, p_{n})^{∗∗})/(N^{∘∘} + l_{2}(X, p_{n})), and for every z ∈ X^{∗∗} we have

k̄(z + X) = k^{∗∗}z + (N^{∘∘} + l_{2}(X, p_{n})) = (z, 0, 0, …) + (N^{∘∘} + l_{2}(X, p_{n})).

Hence k̄ is the first term in the decomposition of T̄, and proceeding as in the last theorem, we can show that Ū is the second term.

We finish the section showing some additional properties of j and k. Recall that T: X → Y is tauberian if T^{∗∗^{−1}}Y = X [8], and it is cotauberian if T^{∗} is tauberian [10]. We note that T is tauberian if and only if T̄ is injective [12], and T is cotauberian if and only if T̄ has dense range [10].

1.7. Proposition. Suppose jUk is the decomposition of T. Then

(a) j is tauberian injective,

(b) k is cotauberian with dense range.

Proof. It is enough to note that j ∈ L(F, Y) is injective, k ∈ L(X, E) has dense range, and j̄Ūk̄ is the decomposition of T̄.

2. Applications

In this section we apply the decomposition to obtain some results for operator ideals [9]. In particular we prove the interpolation property (stronger than the factorization property) for some operator ideals.

2.1. Proposition. Let n be a positive integer.

(a) ∥kx∥ ≤ 2^{n}∥T x∥ + 2^{−n}∥x∥ for every x ∈ X.

(b) jB_{F} ⊂ 2^{n}T B_{X} + 2^{−n}B_{Y} .

Proof. (a) It is enough to note that

(x, 0, …, 0, 0, …) = (0, 0, …, x, 0, …) + (x, 0, …, −x, 0, …)

with (x, 0, …, −x, 0, …) ∈ N.

(b) Clearly (y, y, y, …) ∈ B_{F} implies q_{n}(y) < 1 for every n; i.e., y ∈ 2^{n}T B_{X} + 2^{−n}B_{Y} for every n.

Given operator ideals 𝒜 and ℬ, the product 𝒜 ∘ ℬ is an operator ideal defined by

𝒜 ∘ ℬ(X, Y) := {T ∈ L(X, Y)/T = AB for some A ∈ 𝒜 , B ∈ ℬ}.

2.2. Proposition. Let 𝒜 , ℬ be closed operator ideals.

(a) If 𝒜 is injective and T ∈ 𝒜 , then k ∈ 𝒜 .

(b) If ℬ is surjective and T ∈ ℬ, then j ∈ ℬ.

(c) If 𝒜 is injective and ℬ is surjective, then ℬ ∩ 𝒜 = ℬ ∘ 𝒜 .

Proof. (a) and (b) can be derived from the last proposition, using the characterization of the closed injective hull of an operator ideal in [7; 20.7.3], and the corresponding characterization of the closed surjective hull, respectively.

(c) is a consequence of (a) and (b):

If T ∈ 𝒜 ∩ ℬ then j ∈ B and k ∈ 𝒜; hence jUk = T ∈ ℬ ∘ 𝒜 ,

and the converse implication is evident. This part was proved in [6].

We shall consider now for operator ideals a more restrictive property than the factorization property.

2.3. Definition. An operator ideal 𝒜 has the interpolation property if the identity I_{F} of the intermediate space in the DFJP factorization of T belongs to 𝒜 when T ∈ 𝒜 .

The decomposition can be applied to show that some operator ideals have the interpolation property, as in the following result, proved in [6] using real interpolation techniques. The proof we shall give is more elementary.

2.4. Theorem. Let 𝒜 be an injective and surjective operator ideal verifying the Σ_{2} -condition: For all sequences of Banach spaces (X_{n}) and (Y_{n}), an operator S: l_{2}(X_{n}) → l_{2}(Y_{n}) belongs to 𝒜 when the component operators S_{mn}: X_{n} → Y_{m} in the matricial representation of S belong to 𝒜 .

Then 𝒜 has the interpolation property.

Proof. Let Q and J denote the quotient map onto l_{2}(X, p_{n})/N and the injection of F into l_{2}(Y, q_{n}) respectively, and suppose jUk is the decomposition of T.

An easy computation shows that the components of the matricial representation of the operator JUQ: l_{2}(X, p_{n}) → l_{2}(Y, q_{n}) coincide with T: (X, p_{n}) → (Y, q_{m}); then JUQ ∈ 𝒜 , because 𝒜 satisfies the Σ_{2} -condition; hence U ∈ 𝒜 , since 𝒜 is injective and surjective.

Before continuing, we present some examples.

2.5. Examples. The following operator ideals have the interpolation property.

(a) Operators with finite dimensional range.

(b) Operators with separable range.

(c) Weakly compact operators.

(d) Rosenthal operators.

(e) Banach–Saks operators.

(f) Decomposing operators.

Proof. (a) and (b) are immediate. (c), (d), (e) and (f) follow Theorem 2.4 (see [6]).

As an application of the first part, we present two procedures of construction of operator ideals which preserve the interpolation property.

2.6. Theorem. Let 𝒜 be an operator ideal with the interpolation property. The operator ideals defined by

𝒜^{d}(X, Y) := {T ∈ L(X, Y)/T^{∗} ∈ 𝒜 (Y^{∗}, X^{∗})}

and

𝒜^{co}(X, Y) := {T ∈ L(X, Y)/T̄ ∈ 𝒜 (X^{∗∗}/X, Y^{∗∗}/Y)}

have the interpolation property.

Proof. It is easy to verify that 𝒜^{d} and 𝒜^{co} are operator ideals, and the interpolation property follows from Theorems 1.5 and 1.6, respectively.

We derive some consequences.

Dual results of factorization for operators 11

2.7. Corollary. A quasi-weakly compact operator T; i.e., an operator such that T̄ has finite dimensional range, factors through a quasi-reflexive space.

This is the main result in [1].

2.8. Corollary. Let T ∈ L(X, Y). T^{∗} is Rosenthal if and only if T factors through a Banach space containing no subspaces isomorphic to l_{1} and no quotients isomorphic to c_{0} .

Proof. It follows from the interpolation property of Rosenthal operators, Theorem 1.5 and the fact [4] that E^{∗} contains no copies of l_{1} if and only if E contains no copies of l_{1} and no quotients isomorphic to c_{0} .

2.9. Corollary. Let T ∈ L(X, Y). T̄ has separable range if and only it is T = S + A, with A, S ∈ L(X, Y), R(S^{∗∗}) separable and A weakly compact.

Proof. Since T̄ has separable range, the intermediate space E^{∗∗}/E is separable. Then E is isomorphic to E_{1} × E_{2} , with E_{1} reflexive and E_{2}^{∗∗} separable [11]. Now denoting by P the projection onto E_{1} along E_{2} , it is enough to take A := jUP k and S := jU(I − P)k, where jUk denotes the decomposition of T.

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[3] Davis, W.J., T. Figiel, W.B. Johnson, and A. Pelczynski: Factoring weakly compact operators. - J. Funct. Anal. 17, 1974, 311–327.

[4] Gonzalez, M., and V.M. Onieva: Lifting results for sequences in Banach spaces. - Math. Proc. Camb. Phil. Soc. 105, 1989, 117–121.

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[6] Heinrich, S.: Closed operator ideals and interpolation. - J. Funct. Anal. 35, 1980, 397– 411.

[7] Jarchow, H.: Locally convex spaces. - Teubner, Stuttgart, 1981.

[8] Kalton, N., and A. Wilansky: Tauberian operators in Banach spaces. - Proc. Amer. Math. Soc. 57, 1976, 251–255.

[9] Pietsch, A.: Operator ideals. - North-Holland, Amsterdam, 1978.

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Received 2 May 1991