Lobachevskii Journal of Mathematics Vol. 9, 2001, 15 – 27 \begin{matrix}c \\ ◯ \end{matrix}Eduardo Brandani da Silva, Dicesar L. Fernandez

Eduardo Brandani da Silva, Dicesar L. Fernandez

INTERPOLATION SPACES WITH FUNCTION PARAMETER AND MEASURES OF NON-COMPACTNESS

(submitted by D. Kh. Mustari)

Abstract. The behaviour of the measure of non-compactness of an operator acting on interpolation spaces, with function parameter and having one-sided hypothesis is studied. We obtain as corollaries a theorem of Cwikel type and a result to the radius of the essential spectrum of the interpolated operator.

1. Introduction

One of subject in interpolation theory is the study of the behaviour in the interpolated spaces, of operators acting in the extreme spaces having special properties. The compactness is one of the main properties. Many authors attacked this question, for instance J. L. Lions and J. Peetre [10], A. Persson [12], K. Hayakawa [9], F. Cobos, D. E. Edmunds and J. B. Potter [3], F. Cobos and D. L. Fernandez [4], F. Cobos, T. Khun and T. Schonbek [6], M. Cwikel [7], among others.

In view of all these works on compact operators, a more general question arises: what can we say about the behaviour of non-compact linear operators under interpolation? A powerful tool to analyse non-compact operators is the measure of non-compactness. The concerned problem is thus to give estimates to the measure of non-compactness of a operator acting in the interpolated space, in function of measures of operator acting in the extreme spaces.

16 EDUARDO BRANDANI DA SILVA, DICESAR L. FERNANDEZ

The first results in this way for abstract Banach spaces was obtained by M. F. Teixeira and D. E. Edmunds in [14]. They present results of Lions-Peetre and Persson type using interpolated spaces by the θ-method. In [13] we got some extensions of these results using either approximation or inclusion hypoteses on the extreme spaces, beside one-sided compactness for the operators and interpolation spaces with function parameter. More recently, F. Cobos, P. Fernandez-Martínez and A. Martínez in the paper [5] obtained a result without approximation hypothesys for the θ-method. In this paper we present a theorem for measures of non-compactness of operators acting in interpolated spaces with function parameter but without extra hypotheses on the extreme spaces. As a consequence of our main result, we obtain a compactness theorem of Cwikel type, just assuming compactness only in one of the extreme spaces. Moreover, as another application, we get a result to the radius of the essential spectrum of an operator.

1991 Mathematical Subject Classification. 46M35, 47B05.

2. Interpolation with Function Parameter

We begin with some facts on Interpolation Theory and function parameters which will be needed throughout this paper. We refer the reader to [1] and [8] for a more detailled exposition.

A function parameter is a continuous and positive function on R^{+}. We say that a function parameter ρ belongs to the class P if it satisfies

ρ(s) ≤ max{1, s/t}ρ(t)

for all s, t in R^{+}. Then it follows that ρ(t) is non-decreasing and ρ(t)/t is non-increasing. Besides, we say that a function parameter ρ ∈ P belongs to the class P^{+−} if it satisfies

\overline{ρ}(t) = sup_{s>0} \frac{ρ(st)}{ρ(s)} = o(max\{1, t\}). (1)

From definition (1) we have \overline{ρ} is a submultiplicative function and \overline{ρ}(t)ρ(s) ≤ ρ(st) for all s, t ∈ R^{+}.

A pair of Banach spaces {E_{0}, E_{1}} is called a Banach couple if E_{0} and E_{1} are continuously embedded in some Hausdorff Linear Topological Space E. Then their intersection E_{0} ∩ E_{1} and sum E_{0} + E_{1} become Banach spaces when endowed with the norms

|| x ||_{E_{0}∩E_{1}} = max{||x||_{E_{0}}, ||x||_{E_{1}}}, || x ||_{E_{0}+E_{1}} = inf_{x=x_{0}+x_{1}}{||x_{0}||_{E_{0}} + ||x_{1}||_{E_{1}}},

respectively (see [1] and [2]).

A Banach space E is called an intermediate space with respect to a Banach couple {E_{0}, E_{1}} if E_{0}∩E_{1} ↪ E ↪ E_{0}+E_{1} (where ↪ denotes bounded embeddings). Let {E_{0}, E_{1}} and {F_{0}, F_{1}} be Banach couples and let L({E_{0}, E_{1}}, {F_{0}, F_{1}}) be the family of all linear maps T : E_{0}+E_{1} → F_{0}+F_{1} such that T |_{E_{k}} is bounded from E_{k} to F_{k}, k = 0, 1. If E and F are intermediate spaces with respect to {E_{0}, E_{1}} and {F_{0}, F_{1}} respectively and ρ ∈ P^{+−}, then we say that E and F are interpolation spaces of type ρ if

||T ||_{L(E,F)} ≤ C_{ρ}||T ||_{0} \overline{ρ}(\frac{||T ||_{1}}{||T ||_{0}} ), (2)

for all T ∈ L({E_{0}, E_{1}}, {F_{0}, F_{1}}), where ||T ||_{k} = ||T ||_{L(E_{k},F_{k})}, (k = 0, 1) and C_{ρ} > 0 is a constant.

Now we define the two main interpolation spaces of type ρ. Let {E_{0}, E_{1}} be a Banach couple and consider the functionals J and K defined respectively by

J(t, x) = J(t, x; IE) = max{||x||_{E_{0}}, t ||x||_{E_{1}}}, x ∈ E_{0} ∩ E_{1},

K(t, x) = K(t, x; IE) = inf_{x=x_{0}+x_{1}}{||x_{0}||_{E_{0}} + t ||x_{1}||_{E_{1}}}.

Given ρ ∈ P^{+−} and 1 ≤ q ≤ +∞, the space (E_{0}, E_{1})_{ρ,q,K} consists of all x ∈ E_{0} + E_{1} whose norm ||x||_{ρ,q;K} = ||( ρ(2^{n})^{−1}K(2^{n}, x; E) )_{n∈Z}||_{ℓ^{q}(Z)} is finite, whereas the space (E_{0}, E_{1})_{ρ,q;J}, consists of all x ∈ E_{0} + E_{1}, having a representation x = ∑_{n=−∞}^{∞} u_{n}, where (u_{n}) ∈ E_{0} ∩ E_{1} converges in E_{0} + E_{1} and whose norm

||x||_{ρ,q;J} = inf{||( ρ(2^{n})^{−1}J(2^{n}, u_{n}; IE) )_{n∈ZZ}||_{ℓ^{q}(Z)}}

is finite, where the infimum is taken over all representation x = ∑ u_{n}.

For 1 ≤ q ≤ +∞ and ρ ∈ P^{+−}, the Equivalence Theorem between the J and K method holds, that is, (E_{0}, E_{1})_{ρ,q;J} = (E_{0}, E_{1})_{ρ,q;K}, and using the results of [1] and [2] we get

||x||_{ρ,q,K} ≤ \frac{2 − \overline{ρ}(2)\overline{ρ}(2^{−1})}{2 − \overline{ρ}(2) − 2\overline{ρ}(2^{−1}) + \overline{ρ}(2)\overline{ρ}(2^{−1})} ||x||_{ρ,q,J}

for all x ∈ (E_{0}, E_{1})_{ρ,q,J}. Thus, for the interpolated spaces in (2), we have

||T ||_{L(E,F)} ≤ C_{ρ} ||T ||_{0} \overline{ρ}(\frac{||T ||_{1}}{||T ||_{0}} ) ,

where C_{ρ} ≤ \frac{\overline{ρ}(2)(2−\overline{ρ}(2)\overline{ρ}(2^{−1}))}{2−\overline{ρ}(2)−2\overline{ρ}(2^{−1})+\overline{ρ}(2)\overline{ρ}(2^{−1})}.

We shall also consider the more general classes J(ρ) and K(ρ). Let E be an intermediate space with respect to a Banach couple {E_{0}, E_{1}}. We say that E is an intermediate space of class J_{ρ}(E_{0}, E_{1}) if the embedding (E_{0}, E_{1})_{ρ,1;J} ↪ E holds, and we say that E is an intermediate space of class K_{ρ}(E_{0}, E_{1}) if the embedding E ↪ (E_{0}, E_{1})_{ρ,∞;K} holds. We note that E belongs to the class J_{ρ}(E_{0}, E_{1}) if and only if for all x ∈ E_{0} ∩ E_{1} one has

||x||_{E} ≤ C||x||_{0} \overline{ρ}(\frac{||x||_{1}}{||x||_{0}} ).

3. Measures of Non-Compactness

The concept of measure of non-compactness is very useful in some branches of Functional Analisys and Partial Differential Equations. Bellow we present the definition of measures of non-compactness and we list some of its main important properties.

Let E and F be Banach spaces, let B a bounded subset of E and ε > 0. A cover {V_{i}} of B is a ε-cover if diam(V_{i}) ≤ ε, for all i. The measure of non-compactness of B is defined by

ψ_{E}(B) = inf{ε > 0 : there is a finite ε-cover of B}.

A cover of B by balls {B_{i}} with radius ≤ ε is said to be a ε-ball cover of B. The ball measure of non-compactness of B is defined by

ψ̃_{E}(B) = inf{ε > 0 : there is finite ε-ball cover of B}.

Let E and F be Banach spaces, T ∈ L(E, F) and k ∈ R, k ≥ 0. Then T is called to be a k-set contraction if and only if

ψ_{F} (T (B)) ≤ k ψ_{E}(B)

for all bounded sets B ⊂ E. The measure of non-compactness β(T) of T is defined by

β(T) = min{k : T is a k-set contraction }.

The k-ball contractions and the ball measure of non-compactness of T, β̃(T), are defined analogously, by using the ball measures of non-compactness ψ̃_{E} and ψ̃_{F}.

Let A ⊂ E be a bounded subset and T, T_{1}, T_{2} ∈ L(E, F), then we have the following properties: (i) ψ_{E}(A) = 0 if and only if \overline{A} is compact. (ii) ψ_{E}(A) = ψ_{E}(\overline{A}). (iii) ψ̃_{E}(A) ≤ ψ_{E}(A) ≤ 2ψ̃_{E}(A). (iv) If dim(E) = ∞, then ψ̃_{E}(U_{E}) = 1, where U_{E} is the unitary ball in E. (v) T is compact if and only if β(T) = 0. (vi) β(T) ≤ ||T ||_{L(E,F)}. (vii) β(T_{1} + T_{2}) ≤ β(T_{1}) + β(T_{2}). (viii) β̃(T) = ψ̃_{F} (T (U_{E})). (ix) \frac{1}{2}β(T) ≤ β̃(T) ≤ 2β(T). (x) If S ∈ L(F, G), where G is a Banach spaces, then β(ST) ≤ β(S) β(T). (xi) If T ∈ L(X) = L(X, X), then the radius of the essential spectrum of T, r_{e}(T) is given by

r_{e}(T) = lim_{n→∞}(β̃(T^{n}))^{1/n}.

For more references and proofs of (i) - (x) see [13] and [14]. For the proof of (xi) we refer to [11].

4. The One-Sided Theorem

Let us define some operators and auxiliar spaces that will be needed in what is to follows.

Let {E_{0}, E_{1}} and {F_{0}, F_{1}} be Banach couples. For m ∈ Z let G_{m} = E_{0} ∩ 2^{m}E_{1} and consider the spaces ℓ_{0}^{1}(G_{m}) and ℓ_{1}^{1}(G_{m}), where (x_{n}) ∈ ℓ_{k}^{1}(G_{m}) if

||(x_{n})||_{ℓ_{k}^{1}(G_{m})} = ∑_{n=−∞}^{∞} 2^{−nk}J(2^{n}, x_{n}) < +∞ (k = 0, 1).

Given (x_{n}) ∈ ℓ_{k}^{1}(G_{m}) we define the operator σ by

σ((x_{n})) = ∑_{n=−∞}^{∞} x_{n}.

Hence σ : ℓ_{k}^{1}(G_{m}) → E_{k} (k = 0, 1). For T ∈ L({E_{0}, E_{1}}, {F_{0}, F_{1}}) let T̂ be the operator defined by

T̂ = T ∘ σ : ℓ_{k}^{1}(G_{m}) → F_{k}.

For N ∈ N, we also define P_{N} : ℓ_{k}^{1}(G_{m}) → ℓ_{k}^{1}(G_{m}) (k = 0, 1) by

P_{N}((x_{n})) = (⋯, 0, 0, x_{−N}, x_{−N+1}, ⋯, x_{0}, ⋯, x_{N−1}, x_{N}, 0, 0, ⋯)

so that we have ||P_{N}||_{L(ℓ_{k}^{1}(G_{m}),ℓ_{k}^{1}(G_{m}))} = 1.

Beside the spaces ℓ_{k}^{1}(G_{m}), we will be needed the more general spaces ℓ_{ρ}^{q}(Q_{m}) which are defined as follows.

Let Q be a linear space and let (|| · ||_{n})_{n∈Z} be a sequence of norms on Q. For each n ∈ Z, we shall denote by Q_{n} the space Q equiped with the norm || · ||_{n}: Q_{n} = (Q, || · ||_{n}). Let ρ be any function parameter and 1 ≤ q ≤ ∞. We shall denote by ℓ_{ρ}^{q}(Q_{n}) the linear space of all sequences (a_{n})_{n∈Z}, in Q, such that

|||(a_{n})|||_{ρ,q} = ||(a_{n})_{n∈ZZ}||_{ℓ_{ρ}^{q}(Q_{n})} =[ ∑_{n∈ZZ}[ρ(2^{−n})||a_{n}||_{n}]^{q}]^{1/q} < +∞.

The functional ||| · |||_{ρ,q} is a norm on ℓ_{ρ}^{q}(Q_{n}).

4.1 Lemma. We have, for the above norm that

(ℓ_{0}^{q}(Q_{m}), ℓ_{1}^{q}(Q_{m}))_{ρ,q} = ℓ_{γ}^{q}(Q_{m}), 1 ≤ q ≤ ∞.

where γ(t) = 1/ρ(t^{−1}).

This lemma is a slightly weaker version of a result contained in [8] and [13]. Now we are ready to establish a fundamental Lemma for this work.

4.2 Lemma. Let {E_{0}, E_{1}}, {F_{0}, F_{1}} and {ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m})} be Banach couples, where G_{m} is defined as above. Given T ∈ L({E_{0}, E_{1}}, {F_{0}, F_{1}}) and ε > 0, there exists T̂ ∈ L({ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m})}, {F_{0}, F_{1}}) and an operator P_{N} ∈ L({ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m})}, {ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m})}) such that, for k = 0, 1, we have

||T̂ − T̂P_{N}||_{L(ℓ_{k}^{1}(G_{m}),F_{k})} ≤ 4 β̃(T_{k}) + ε,

where β̃(T_{k}) = β̃(T_{E_{k},F_{k}}).

Proof: Let us denote by U_{E} and \overline{U}_{E}, respectively the open unitary ball and the closed unitary ball of a Banach space E and by ℓ_{k}^{1} the spaces ℓ_{k}^{1}(G_{m}). Given x̂ ∈ \overline{U}_{ℓ_{k}^{1}}, we have x̂ = (x_{n}) with x_{n} ∈ E_{0} ∩ 2^{n}E_{1} and ||(x_{n})||_{ℓ_{k}^{1}} = ∑_{n=−∞}^{∞} 2^{−nk}J(2^{n}, x_{n}) ≤ 1. By definition of the norm in ℓ_{k}^{1} we obtain

||σ(x̂)||_{E_{k}} ≤ ∑_{n=−∞}^{+∞} ||x_{n}||_{E_{k}} ≤ ∑_{n=−∞}^{+∞} 2^{−nk} max{||x_{n}||_{E_{0}}, 2^{nk}||x_{n}||_{E_{1}}} = ||x̂||_{ℓ_{k}^{1}} ≤ 1,

for k = 0, 1. Therefore, for σ : ℓ_{k}^{1} → E_{k}, k = 0, 1, we have ||σ||_{L(ℓ_{k}^{1},E_{k})} ≤ 1.

By definition of σ, and by properties (iii) and (viii) of section 3, we have σ(\overline{U}_{ℓ_{k}^{1}}) ⊂ \overline{U}_{E_{k}} and

ψ_{F_{k}}(T (\overline{U}_{E_{k}})) ≤ 2 β̃(T_{k}).

From the definition of the measure ψ, given ε > 0, for each k there exists a finite cover (V^{∘}_{i,k}) of T (\overline{U}_{E_{k}}), where V_{i,k} are closed balls, such that

diam(V_{i,k}) ≤ 2 β̃(T_{k}) + \frac{ε}{3}.

We have \overline{U}_{E_{k}} ⊂ ∪_{i=1}^{n_{k}}T^{−1}(V_{i,k}), and it follows that

\overline{U}_{ℓ_{k}^{1}} ⊂ σ^{−1}(\overline{U}_{E_{k}}) ⊂ ∪_{i=1}^{n_{k}}σ^{−1}(T^{−1}(V_{i,k})).

Hence

diam(T̂(T̂^{−1}(V_{i,k}))) ≤ diam(V_{i,k}) ≤ 2β̃(T_{k}) + \frac{ε}{3}, (3)

where T̂ = T σ. We can suppose T̂^{−1}(V^{∘}_{i,k}) ∩ \overline{U}_{ℓ_{k}^{1}} ≠ ∅ for all i, and for each one we can choose ŷ_{i,k} ∈ T̂^{−1}(V^{∘}_{i,k}) ∩ \overline{U}_{ℓ_{k}^{1}}. Since we have T̂ŷ_{i,k} ∈ V_{i,k}^{∘}, there exists an ε′_{i} > 0 such that B( T̂ŷ_{i,k}, ε′_{i}) ⊂ V_{i,k}. Let ε′ = min{ε′_{1}, …, ε′_{n_{k}}, ε}. Thus, B(T̂ŷ_{i}, ε′) ⊂ V_{i,k}, ∀ i = 1, ..., n_{k}, and, since T̂ is bounded , setting δ = \frac{ε}{||T ||} we have

T̂(B(ŷ_{i,k}, δ)) ⊂ B(T̂ŷ_{i,k}, ε′) ⊂ V_{i,k} .

Since we have finite i, there exists an N, large enough, such that

P_{N} ŷ_{i,k} ∈ B(ŷ_{i,k}, \frac{δ}{3}), (4)

for all i. Therefore, we have a finite quantity of elements ŷ_{i,k}, (k = 0, 1), and given ε > 0, there is an N ∈ N such that

||T̂ŷ_{i,k} − T̂P_{N} ŷ_{i,k}||_{F_{k}} ≤ \frac{ε}{3} (5)

for all ŷ_{i,k}.

Now, we claim that P_{N}^{−1}(T̂^{−1}(V_{i,k})) ≠ ∅ for all i, and k = 0, 1. Indeed, for x̂_{i,k} ∈ B(ŷ_{i,k}, \frac{δ}{2}), we have

||P_{N} x̂_{i,k} − ŷ_{i,k}||_{ℓ_{k}^{1}} = ||P_{N} x̂_{i,k} − P_{N} ŷ_{i,k} + P_{N} ŷ_{i,k} − ŷ_{i,k}||_{ℓ_{k}^{1}}
≤ ||P_{N} x̂_{i,k} − P_{N} ŷ_{i,k}||_{ℓ_{k}^{1}} + ||P_{N} ŷ_{i,k} − ŷ_{i,k}||_{ℓ_{k}^{1}}
≤ ||P_{N}||_{L(ℓ_{k}^{1},ℓ_{k}^{1})}||x̂_{i,k} − ŷ_{i,k}||_{ℓ_{k}^{1}} + ||P_{N} ŷ_{i,k} − ŷ_{i,k}||_{ℓ_{k}^{1}}
≤ \frac{δ}{2} + \frac{δ}{2} = δ.

Hence, P_{N} x̂_{i,k} ∈ B(ŷ_{i_{,k}}, δ), then T̂P_{N} x̂_{i,k} ∈ T̂(B(ŷ_{i_{,k}}, δ)) ⊂ B(T̂ŷ_{i,k}, ε′) ⊂ V_{i,k} which implies that x̂_{i,k} ∈ P_{N}^{−1}(T̂^{−1}(V_{i,k})). Then, for all x̂ ∈ \overline{U}_{ℓ_{k}^{1}}, there exists i and ŷ_{i,k} such that T̂x̂, T̂ŷ_{i,k} ∈ V_{i,k}. Since P_{N}(\overline{U}_{ℓ_{k}^{1}}) ⊂ \overline{U}_{ℓ_{k}^{1}(G_{m})}, (k = 0, 1) we obtain

T̂(P_{N}(\overline{U}_{ℓ_{k}^{1}})) ⊂ T̂(\overline{U}_{ℓ_{k}^{1}}) ⊂ ⋃_{i=1}^{n_{k}} V_{i,k},

which implies

\overline{U}_{ℓ_{k}^{1}} ⊂ ⋃_{i=1}^{n_{k}} P_{N}^{−1}(T̂^{−1}(V_{i,k})).

Thus,

sup_{x̂∈P_{N}^{−1}(T̂^{−1}(V_{i,k}))∩\overline{U}_{ℓ^{1}_{k}}} ||T̂P_{N} x̂ − T̂P_{N} ŷ_{i}||_{F_{k}} ≤ sup_{x̂∈P_{N}^{−1}(T̂^{−1}(V_{i,k}))} ||T̂P_{N} x̂ − T̂P_{N} ŷ_{i}||_{F_{k}}

≤ sup_{x̂,ŷ∈P_{N}^{−1}(T̂^{−1}(V_{i,k}))} ||T̂P_{N} x̂ − T̂P_{N} ŷ||_{F_{k}}
≤ sup_{x̂,ŷ∈T̂^{−1}(V_{i,k})} ||T̂x̂ − T̂ŷ||_{F_{k}}
= diam(T̂(T̂^{−1}(V_{i,k}))) ≤ 2 β̃(T_{k}) + \frac{ε}{3}

and we have

||T̂P_{N} x̂ − T̂P_{N} ŷ_{i}||_{F_{k}} ≤ 2 β̃(T_{k}) + \frac{ε}{3} (6)

for all x̂ ∈ U_{ℓ_{k}^{1}} and some ŷ_{i}. Therefore, given x̂ ∈ \overline{U}_{ℓ_{k}^{1}}, by (3), (5) and (6), we have a ŷ_{i,k} such that

||(T̂ − T̂P_{N})x̂||_{F_{k}} = ||T̂x̂ − T̂P_{N} x̂||_{F_{k}}
= ||T̂x̂ − T̂P_{N} x̂ − T̂ŷ_{i,k} + T̂ŷ_{i,k} − T̂P_{N} ŷ_{i,k} + T̂P_{N} ŷ_{i,k}||_{F_{k}}
≤ ||T̂x̂ − T̂ŷ_{i,k}||_{F_{k}} + ||T̂ŷ_{i,k} − T̂P_{N} ŷ_{i,k}||_{F_{k}} + ||T̂P_{N} ŷ_{i,k} − T̂P_{N} x̂||_{F_{k}}
≤ 2 β̃(T_{k}) + \frac{ε}{3} + \frac{ε}{3} + 2 β̃(T_{k}) + \frac{ε}{3}
≤ 4 β̃(T_{k}) + ε

which is the desired inequality.

Using an idea from [5] we can proof the following Lemma.

4.3 Lemma. Let {E_{0}, E_{1}} and {F_{0}, F_{1}} be Banach couples. Let E = (E_{0}, E_{1})_{ρ,q} and F = (F_{0}, F_{1})_{ρ,q} where ρ ∈ P^{+ −}and 1 ≤ q ≤ ∞. Given the operator T σP_{N} ∈ L((ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m}))_{ρ,q}, F) defined above we have

β̃(T σP_{N})_{L((ℓ_{0}^{1},ℓ_{1}^{1})_{ρ,q},(F_{0},F_{1})_{ρ,q})} ≤ C_{ρ} \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)}.

Proof: By the Lemma 4.1, we have (ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m}))_{ρ,q} = ℓ_{γ}^{q}(G_{m}), where γ(t) = ρ(t^{−1})^{−1}. By definition of P_{N} : ℓ_{γ}^{q} → ℓ_{γ}^{q} we can indentify P_{N}(ℓ_{γ}^{q}) with R_{q}^{2N+1}, where

R_{q}^{2N+1} = {(a_{−N}, ⋯, a_{N}) ∈ R^{2N+1} : ( ∑_{n=−N}^{N} |a_{n}|^{q})^{1/q} < +∞} .

As the dimension(R_{q}^{2N+1}) = 2N + 1 < +∞, the unit ball U_{R_{q}^{2N+1}} of R_{q}^{2N+1} is compact. Therefore, given an ε-covering of U_{R_{q}^{2N+1}} by balls (B_{k}) with radius(B_{k}) ≤ ε, k ∈ Λ, where Λ is a set of index, there is a finite subcovering (B_{r})_{r=1}^{s} of the U_{R_{q}^{2N+1}}. If {z_{r}}_{r=1}^{s} is the set of the centers of the balls B_{r}, given λ ∈ U_{R_{q}^{2N+1}} we have

min_{1≤r≤s}{||λ − z_{r}||_{R_{q}^{2N+1}}} ≤ ε,

where z_{r} = (z_{r}^{−N}, …, z_{r}^{N}). But, given (x_{n})_{−∞}^{+∞} ∈ ℓ_{γ}^{q}, we have

||P_{N}(x_{n})||_{ℓ_{γ}^{q}} = (∑_{−N}^{N} (ρ(2^{n})^{−1} J(2^{−n}, x_{n}))^{q})^{1/q} ,

then if we set λ = (ρ(2^{−N})^{−1}J(2^{−N}, x_{−N}), …, ρ(2^{N})^{−1}J(2^{N}, x_{N})) we shall have λ ∈ R_{q}^{2N+1}. In the same way, if (x_{n})_{−∞}^{+∞} ∈ U_{ℓ_{γ}^{q}} we have λ = (ρ(2^{n})^{−1}J(2^{n}, x_{n}))_{n=−N}^{N} ∈ U_{R_{q}^{2N+1}}. Therefore, there exists z_{r} such that ||λ − z_{r}||_{R_{q}^{2N+1}} ≤ ε, r ∈ 1, …, s. Thus, for −N ≤ n ≤ N, we have

|ρ(2^{n})^{−1}J(2^{n}, x_{n}) − z_{r}^{n}| ≤ (∑_{−N}^{N} |ρ(2^{n})^{−1}J(2^{n}, x_{n}) − z_{r}^{n}|^{q})^{1/q} ≤ ε,

which implies

ρ(2^{n})^{−1}J(2^{n}, x_{n}) ≤ ε + z_{r}^{n},

and, for v ∈ Z

\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ}(2^{v−1})^{−1}J(2^{n−v+v}, x_{n}) ≤ ε + z_{r}^{n}.

Now, let k_{i} be real numbers such that k_{i} > β̃_{i}(T)+δ, where δ is an arbitrary but fixed positive number, i = 0, 1. This given, let v ∈ Z such that 2^{v−1} < k_{1}/k_{0} ≤ 2^{v}. Then, since k_{1}/k_{0} ≤ 2^{v} and J(t, x) is increasing in t, we have

\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ}(2^{v−1})^{−1}J(2^{n−v} \frac{k_{1}}{k_{0}} , x_{n}) ≤

≤ \overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ}(2^{v−1})^{−1}J(2^{n−v+v}, x_{n}) ≤ ε + z_{r}^{n}

And, since \overline{ρ}(k_{1}/k_{0})^{−1} ≤ \overline{ρ}(2^{v−1})^{−1}, the above inequality yields to

\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ} (\frac{k_{1}}{k_{0}} )^{−1} J(2^{n−v} \frac{k_{1}}{k_{0}} , x_{n}) ≤ ε + z_{r}^{n}.

Therefore,

J (2^{n−v} \frac{k_{1}}{k_{0}} , (\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ} (\frac{k_{1}}{k_{0}} )^{−1}) x_{n}) ≤ ε + z_{r}^{n} (7)

for some r ∈ 1, …, s and all n ∈ −N, …, N.

Given (x_{n})_{n=−∞}^{+∞} ∈ U_{ℓ_{γ}^{q}}, by definition, we have (∑_{−∞}^{+∞} (ρ(2^{n})^{−1}J(2^{n}, x_{n}))^{q})^{1/q} ≤

1. This implies ρ(2^{n})^{−1}J(2^{n}, x_{n}) ≤ 1, what is the same to say that ρ(2^{n})^{−1}x_{n} ∈ U_{E_{0}} and 2^{n}ρ(2^{n})^{−1}x_{n} ∈ U_{E_{1}}. Thus, in (7 )we have

\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ} (\frac{k_{1}}{k_{0}} )^{−1} (ε + z_{r}^{n})^{−1}x_{n} ∈ U_{E_{0}}

and

\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ} (\frac{k_{1}}{k_{0}} )^{−1} 2^{n−v} \frac{k_{1}}{k_{0}} (ε + z_{r}^{n})^{−1}x_{n} ∈ U_{E_{1}}.

Since ψ̃(T (U_{E_{i}})) ≤ k_{i}, there are c_{i,1}, …, c_{i,l_{i}} ∈ F_{i} such that

T (U_{E_{i}}) ⊂ ∪_{j=1}^{l_{i}}(c_{i,j} + k_{i}U_{F_{i}}).

Then, given x ∈ U_{E_{i}}, there exists c_{i,j} such that ||T x − c_{i,j}||_{F_{i}} ≤ k_{i} and this implies

∥T (\overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ} (\frac{k_{1}}{k_{0}} )^{−1} 2^{i(n−v)} (\frac{k_{1}}{k_{0}} )^{i} (ε + z_{r}^{n})^{−1}x_{n}) − c_{i,j}∥_{F_{i}} ≤ k_{i}.

Therefore, we have

||T x_{n} − b_{i,j,n}||_{F_{i}} ≤ k_{i}\overline{ρ}(2)ρ(2^{n−v})\overline{ρ} (\frac{k_{1}}{k_{0}} ) 2^{−i(n−v)} (\frac{k_{1}}{k_{0}} )^{−i} (ε + z_{r}^{n}),

where b_{i,j,n} = M_{i,j}^{−1}c_{i,n} with

M_{i,n} = \overline{ρ}(2)^{−1}ρ(2^{n−v})^{−1}\overline{ρ} (\frac{k_{1}}{k_{0}} )^{−1} 2^{i(n−v)} (\frac{k_{1}}{k_{0}} )^{i} (ε + z_{r}^{n})^{−1}.

Let us set D_{j,g}^{n} = {b_{0,n,j} + M_{0,n}^{−1}U_{F_{0}}} ∩ {b_{1,n,g} + M_{1,n}^{−1}U_{F_{1}}}, j = 1, …, l_{0} and g = 1, …, l_{1}. The D_{j,g}^{n} are not empty, and if d_{n,p} ∈ D_{j,g}^{n} we shall have d_{n,p} ∈ F_{0} ∩ F_{1}. Then, given u = (x_{n})_{n=−∞}^{+∞} ∈ U_{ℓ_{γ}^{q}} there is some r ∈ 1, …, s and some {d_{m,p}}_{n=−N}^{N} such that

J (2^{n−v}, T x_{n} − d_{n,p}) = max_{i=0,1} {2^{i(n−v)} ∥T x_{n} − d_{n,p}∥_{F_{i}}}
= max_{i=0,1} {2^{i(n−v)} ∥T x_{n} − b_{n,l}^{i} + b_{n,l}^{i} − d_{n,p}∥_{F_{i}}}
≤ max {2k_{0}M_{0,n}^{−1}, 2^{n−v}2k_{1}M_{1,n}^{−1}}
= 2k_{0}M_{0,n}^{−1}.

Now, we define D be the set of all sums ∑_{−N}^{N} d_{n,p}, where p ∈ 1, …, w and w is the number of sets D_{j,g}^{n}. Then D is a finite subset of (F_{0}, F_{1})_{ρ,q} such that, for all u ∈ U_{ℓ_{γ}^{q}} there exists some ∑_{−N}^{N} d_{n,p} ∈ D with

∥T σP_{N}u − ∑_{−N}^{N} d_{n,p}∥_{ρ,q,K} = ∥∑_{−N}^{N} (T x_{n} − d_{n,p})∥_{ρ,q,K}

≤ C_{ρ} ∥∑_{−N}^{N} (T x_{n} − d_{n,p})∥_{ρ,q,J}

≤ C_{ρ} (∑_{−N}^{N} (ρ(2^{n−v})^{−1}J(2^{n−v}, T x_{n} − d_{n,p}))^{q})^{1/q}

≤ C_{ρ} (∑_{−N}^{N}(ρ(2^{n−v})^{−1}2k_{0}\overline{ρ}(k_{1}/k_{0})\overline{ρ}(2)ρ(2^{n−v})(ε + z_{r}^{n}))^{q})^{1/q}

≤ 2\overline{ρ}(2)C_{ρ} \frac{\overline{ρ}(1/k_{0})}{1/k_{0}} \overline{ρ}(k_{1})(1 + (2N + 1)^{1/q})ε.

Therefore,

β̃(T σP_{N})_{L((ℓ_{0}^{1},ℓ_{1}^{1})_{ρ,q},(F_{0},F_{1})_{ρ,q})} ≤ β̃(T σP_{N})_{L(ℓ_{γ}^{q},(F_{0},F_{1})_{ρ,q})}
≤ 2\overline{ρ}(2)C_{ρ} \overline{ρ}(β̃(T_{1}) + δ) \frac{\overline{ρ}(1/(β̃(T_{0}) + δ))}{1/(β̃(T_{0}) + δ)}.

Finally, we rename δ to ε, and the result follows.

Now we are be able to proof our main result.

4.4 Theorem. Let {E_{0}, E_{1}} and {F_{0}, F_{1}} be Banach couples. Let E = (E_{0}, E_{1})_{ρ,q} and F = (F_{0}, F_{1})_{ρ,q} where ρ ∈ P^{+−} and 1 ≤ q ≤ ∞. Then, given ε > 0 and T ∈ L({E_{0}, E_{1}}, {F_{0}, F_{1}}) we have

β̃(T_{E,F}) ≤ C_{ρ} \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)}.

Proof: Step 1: Let Q_{m} = G_{m} in the definition of spaces ℓ_{ρ}^{q}. Let P_{N} : ℓ_{k}^{1}(G_{m}) → ℓ_{k}^{1}(G_{m}), σ : ℓ_{k}^{1}(G_{m}) → E_{k} (k = 0, 1) and T̂ = T ∘ σ operators defined as in the Lemma 4.2 and ε > 0. By the Lemma 4.1 we have (ℓ_{0}^{1}(G_{m}), ℓ_{1}^{1}(G_{m}))_{ρ,q} = ℓ_{γ}^{q}(G_{m}) where γ(t) = 1/ρ(t^{−1}). We denote by ℓ_{k}^{1} and ℓ_{γ}^{q} respectively the spaces ℓ_{k}^{1}(G_{m}) and ℓ_{γ}^{q}(G_{m}).

Let T̂ = T̂P_{N} + (T̂ − T̂P_{N}) and let B be a bounded subset of ℓ_{γ}^{q}. Thus we have

ψ̃_{F} (T̂(B)) ≤ ψ̃_{F} (T̂P_{N}(B)) + ψ̃_{F} ((T̂ − T̂P_{N})(B))
≤ β̃(T̂P_{N}) ψ̃_{ℓ_{γ}^{q}}(B) + ||T̂ − T̂P_{N}||_{L(ℓ_{γ}^{q},F)} ψ̃_{ℓ_{γ}^{q}}(B).

By the Lemma 4.3, we have the following estimate for the measure of noncompactness β̃(T̂P_{N}):

β̃((T̂P_{N})_{ℓ_{γ}^{q},F}) ≤ 2\overline{ρ}(2)C_{ρ} \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)}.

Step 2: Since ℓ_{γ}^{q} and F are interpolation spaces of type ρ, in view of Lemma 4.2 and the fact that \frac{ρ(t)}{t} is non-increasing, we have by assuming ε′ = 4 ε that

||T̂ − T̂P_{N}||_{L(ℓ_{γ}^{q},F)} ≤ C_{ρ} ||T̂ − T̂P_{N}||_{0} \overline{ρ}(\frac{||T̂ − T̂P_{N}||_{1}}{||T̂ − T̂P_{N}||_{0}} )

≤ C_{ρ} \overline{ρ}(||T̂ − T̂P_{N}||_{1}) \frac{\overline{ρ}(1/||T̂ − T̂P_{N}||_{0})}{1/||T̂ − T̂P_{N}||_{0}}

≤ C_{ρ} \overline{ρ}(4 β̃(T_{1}) + ε′) \frac{\overline{ρ}(1/(4 β̃(T_{0}) + ε′))}{1/(4 β̃(T_{0}) + ε′)}
≤ C_{ρ} \overline{ρ}(4) \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/4)}{1/4} \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)}
≤ \frac{\overline{ρ}(4)\overline{ρ}(1/4)}{1/4}C_{ρ} \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)}.

Step 3: Putting together the results of the steps 1 and 2 we have

ψ̃_{F} (T̂(B)) ≤ β̃(T̂P_{N}) ψ̃_{ℓ_{γ}^{q}}(B) + ||T̂ − T̂P_{N}||_{L(ℓ_{γ}^{q},F)} ψ̃_{ℓ_{γ}^{q}}(B)
≤ C′_{ρ} \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)} ψ̃_{ℓ_{γ}^{q}}(B),

where C′_{ρ} = (2\overline{ρ}(2) + 4\overline{ρ}(4)\overline{ρ}(1/4))C_{ρ}.

Taking B = U_{ℓ_{γ}^{q}} in the above inequality we obtain

ψ̃_{F} (T̂(U_{ℓ_{γ}^{q}})) ≤ C′_{ρ} \overline{ρ}(β̃(T_{1}) + ε) \frac{\overline{ρ}(1/(β̃(T_{0}) + ε))}{1/(β̃(T_{0}) + ε)}

since ψ̃_{ℓ_{γ}^{q}}(U_{ℓ_{γ}^{q}}) = 1.

Step 4: To finish the theorem we claim that σ(U_{ℓ_{γ}^{q}}) = U_{E}. Indeed, if x̂ = (x_{n}) ∈ U_{ℓ_{γ}^{q}} then

||x̂||_{ℓ_{γ}^{q}} = ( ∑_{n=−∞}^{+∞}(γ(2^{−n}) J(2^{n}, x_{n}))^{q})^{1/q} < 1 ,

and since that γ(2^{−n}) = ρ(2^{n})^{−1} and σ(x̂) = ∑ x_{n} we obtain

||σ(x̂)||_{E} = inf ; σ(x̂)=∑ u_{n} ||ρ(2^{n})^{−1} J(2^{n}, u_{n})||_{ℓ^{q}(Z)}
≤ ||ρ(2^{n})^{−1} J(2^{n}, x_{n})||_{ℓ^{q}(Z)}
= ( ∑_{n=−∞}^{+∞}(γ(2^{−n}) J(2^{n}, x_{n}))^{q})^{1/p}
= ||x̂||_{ℓ_{γ}^{q}} < 1

and this implies that σ(U_{ℓ_{γ}^{q}}) ⊂ U_{E}. On the other hand, if x ∈ U_{E}, we have

||x||_{E} = inf ; x=∑ u_{n} ||ρ(2^{n})^{−1} J(2^{n}, u_{n})||_{ℓ^{q}(Z)} < 1,

where u_{n} ∈ E_{0} ∩ E_{1} and thus u_{n} ∈ E_{0} ∩ 2^{n}E_{1}. Hence by the definition of the norm ||.||_{E} we can choose (u′_{n}) ∈ E_{0} ∩ 2^{n} E_{1} such that x = ∑ u′_{n} and ||ρ(2^{n})^{−1} J(2^{n}, u′_{n})||_{ℓ^{q}(Z)} < 1, so that (u′_{n}) ∈ U_{ℓ_{γ}^{q}}, which implies U_{E} ⊂ σ(U_{ℓ_{γ}^{q}}) and this proves the claim. Then we have by property (viii) of the section 3,

ψ̃_{F} (T̂(U_{ℓ_{γ}^{q}})) = ψ̃_{F} (T (σ(U_{ℓ_{γ}^{q}}))) = ψ̃_{F} (T (U_{E})) = β̃(T_{E,F})

and the result follows.

Let us note that, from this theorem we obtain a new proof, to the ρ-method, of a result obtained by Cwikel in [5] for compact operators.

4.5 Corollary. Let {E_{0}, E_{1}} and {F_{0}, F_{1}} be Banach couples. Let E = (E_{0}, E_{1})_{ρ,q}, F = (F_{0}, F_{1})_{ρ,q}, where ρ ∈ P^{+−}, 1 ≤ q ≤ ∞. Then if T ∈ L({E_{0}, E_{1}}, {F_{0}, F_{1}}) is such that T : E_{0} → F_{0} is compact, we have that T : E → F is compact.

4.6. Corollary. Let {E_{0}, E_{1}} be a Banach couple and let E = (E_{0}, E_{1})_{ρ,q} where ρ ∈ P^{+−} and 1 ≤ q ≤ ∞. If T ∈ L({E_{0}, E_{1}}, {E_{0}, E_{1}}) then for the radius of essential spectrum r_{e}, we have

r_{e}(T_{E,E}) ≤ \overline{ρ}(r_{e}(T_{E_{1},E_{1}})) \frac{\overline{ρ}(1/(r_{e}(T_{E_{0},E_{0}})))}{1/r_{e}(T_{E_{0},E_{0}})},

where we are supposing r_{e}(T_{E_{0},E_{0}}) > 0.

Proof: By the Theorem 4.3, with E_{0} = F_{0}, E_{1} = F_{1}, T = T^{n} for any n ∈ N and taking ε → 0 we have

β̃(T_{E,E}^{n}) ≤ C′_{ρ} \overline{ρ}(β̃(T_{1}^{n})) \frac{\overline{ρ}(1/(β̃(T_{0}^{n})))}{1/β̃(T_{0}^{n})}.

Now, let us see that \overline{ρ}(β̃(T_{1}^{n}))^{1/n} = \overline{ρ}((β̃(T_{1}^{n})^{1/n})^{n})^{1/n} ≤ \overline{ρ}(β̃(T_{1}^{n})^{1/n}), and using a similar calculation for \frac{\overline{ρ}(1/(β̃(T_{0}^{n})))}{1/β̃(T_{0}^{n})} we obtain

β̃(T_{E,E}^{n})^{1/n} ≤ C′_{ρ} \overline{ρ}(β̃(T_{1}^{n})^{1/n}) \frac{\overline{ρ}(1/(β̃(T_{0}^{n})^{1/n}))}{1/β̃(T_{0}^{n})^{1/n}}.

Since r_{e}(T) = lim_{n→∞}(β̃(T^{n}))^{1/n} the corollary follow.

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INTERPOLATION SPACES WITH FUNCTION PARAMETER 27

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Departamento de Matemática. Universidade Estadual de Maringá. Av. Colombo, 5790, 87020-900 Maringá - PR - Brazil Instituto de Matematica, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970 Campinas - SP - Brazil. E-mail address: edubrand@bol.com.br

Received November, 6, 2001