The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces

1 Introduction

The formal study of infinite matrices began with the work of Poincaré in 1884 [1], who gave the rigorous bases for the use of infinite matrices for the calculus of the determinants. Poincaré’s results provided Hilbert [2] with the necessary tools to study the eigenvalues of certain integral operators, which can be viewed as infinite matrices operating on specific spaces. This theory enables the study of systems of infinite differential equations such as we can see in the work of Cooke [3]. A walk about the history of infinite matrices can be seen in the excellent article of Bernkopf [4], while the modern approach of this topics and its applications can be found in the article of Shivakumar and Sivakumar [5] and the book of Shivakumar et al. [6].

A complex infinite matrix is denoted by A = ( a i j ) , and it is a function A : N × N → C . As usual, the product of an infinite matrix A by the sequence x = { x ( n ) } , denoted as y = A × x , is the sequence y = { y ( n ) } defined by

whenever the aforementioned series is convergent for all n ∈ N . Thus, if A × x is calculable for all x in a vector space V , then the set

is also a vector space, and we could define the linear operator T : V → W by T A ( x ) = A × x . So far, we can ask for the following:

Given a vector space V , what is the vector space W defined in (1)?.

This last question is related to the well known problem of the multipliers between vector spaces. If the vector spaces V and W are Banach spaces, then we can define the norm of the infinite matrix A as the norm of the operator T A ; that is,

where ∥ x ∥ V denotes the norm of the sequence x ∈ V . In this context, we are interested in finding conditions on the infinite matrix A to the supremum in the right side of (2) will be finite; that is, when is T A : V → W a continuous operator? and, in this case, when is T A : V → W a compact operator? and, how much is the essential norm of T A ?

Recently, interesting works have appeared that answer the questions in the previous paragraph for certain classes of sequence spaces. Infinite matrices that maps a weighted l 1 ( r ) space into another of the same kind l 1 ( s ) were characterized by Williams and Ye [7], they showed that if A is an infinite matrix, r and s are weight sequences (i.e., r ( n ) > 0 for all n ∈ N ), then T A maps l 1 ( r ) into l 1 ( s ) if and only if

and the most general case when T A maps l p ( r ) into l q ( s ) is still an open problem. The infinite matrices that map a l p space into a Cesàro space ces p were studied by Foroutannia and Roopaei in [8]. It is of interest to characterize the infinite matrices that define bounded linear operators between certain spaces of sequences and to study other topological properties such as compactness and closedness of the range.

In this article, we consider the case when A is an infinite diagonal matrix acting on the general setting of Köthe sequence spaces, which include l p ( r ) spaces, Orlicz sequence spaces, Lorentz sequence spaces, and Cesàro sequence spaces among others. Given a sequence u = { u ( n ) } , the infinite matrix diagonal with entries u , denoted by A = diag ( u ) , is the matrix A = ( a i , j ) such that a i , j = u ( i ) for i = j and a i , j = 0 for i ≠ j . In this case, the operator T A = T diag ( u ) is known as the multiplication operator with symbol u , and it is denoted by M u and satisfies that

for all x ∈ V . This operator is well known and continues to be the subject of extensive study by a large number of researchers. In the context of Banach spaces of sequences, we can mention that if V is a Köthe-type sequence space (see the next section for the definition and properties of Köthe sequence spaces), then M u maps V into itself if and only if u ∈ l ∞ (see, for instances, [9]), and more recently, Ramos-Fernández and Salas-Brown [10] have proven that, in this case, M u : V → V is a compact operator if and only if u ∈ c 0 , that is, u n → 0 as n → ∞ . Similar results have been obtained by various authors in different Banach sequence spaces, such as Lorentz sequence spaces [11], Orlicz-Lorentz sequence spaces [12,13], Cesàro sequence spaces [14], Cesàro-Orlicz sequence spaces [15], among others.

In the context of compactness, one of the objectives is to estimate the essential norm of an operator or the non-compactness measure. Given a Banach space ( V , ‖ ⋅ ‖ V ) , the essential norm of a continuous linear operator T : V → V , denoted by ∥ T ∥ e , is defined as the distance to the class K ( V ) of all compact operators on V , that is,

where ‖ T ‖ denotes the operator norm of T . Since K ( V ) is a closed subspace of ℬ ( X ) , the class of all bounded linear operator (with the topology induced by the operator norm), we can see that an operator T ∈ ℬ ( V ) is compact if and only if ‖ T ‖ e = 0 . Recently, interesting works have appeared about the essential norm of multiplication operators acting on Banach sequences spaces. In the case of Lorentz sequence spaces l ( p , q ) , Castillo et al. in [16] proved that the essential norm of the operator M u : l ( p , q ) → l ( p , q ) is given by

which implies the result of Arora et al. in [11]. Ramos-Fernández et al. [17] showed that the aforementioned result is also valid for Cesàro sequence spaces, and more recently, Ramos-Fernández and Salas-Brown [18] have proven that the same is still valid for Orlicz sequence spaces. The main objective of this article is to prove that the result obtained by Castillo et al. in [16] is valid for the more general setting of Köthe-type spaces, which generalize all the results mentioned in this paragraph, and the proof of this aim is given in Section 3. The article is completed in Section 2 where we gather some properties of Köthe sequence spaces.

2 Some remarks about Köthe sequence spaces

In this section, we use the notation a = { a ( k ) } for a sequence of complex number, while ( x n ) will denote a sequence of sequences. With these notations, a Banach space of sequence ( X , ‖ ⋅ ‖ ) is a Köthe sequence space if it satisfies:

1. If a , b are sequences with ∣ a ( k ) ∣ ≤ ∣ b ( k ) ∣ for all k ∈ N and b ∈ X , then a ∈ X and ∥ a ∥ X ≤ ∥ b ∥ X .

2. If A = { n 1 , n 2 , … , n m } is a finite subset of N , then the sequence 1 A defined by

   1 A ( k ) = 1 , k ∈ A , 0 , k ∉ A ,

   belongs to X .

The Köthe dual X ′ of the Köthe sequence space X is a Banach space, which can be identified with the space of all functionals possessing a series representation, that is, c ∈ X ′ if and only if

The space ( X ′ , ‖ ⋅ ‖ X ′ ) is also a Köthe sequence space, and if c ∈ X ′ , then the relation h c : X → C defined by

is a bounded functional on X and ∥ h c ∥ = ∥ c ∥ X ′ . It is known that if X has order continuous norm (i.e., ∥ x n ∥ X → 0 for all sequences ( x n ) ⊂ X such that x n ↓ 0 ), then the dual space X ∗ can be naturally identified with X ′ [25]. In particular, for each n ∈ N , the canonical sequence e n belongs to X ′ , and we have the following useful property. We include its proof for benefit of the reader.

3 Main result

The aim of this section is to obtain an expression for the essential norm of the multiplication operator acting on Köthe sequence spaces, which implies the results obtained by Castillo et al. [16], Ramos-Fernández et al. [17], and Ramos-Fernández and Salas-Brown [18] among others. The essential norm of the bounded operator is given by

where K ( X ) denotes the class of all compact operators defined on X and ∥ T ∥ is the norm of the operator T : X → X . In particular, the operator T : X → X is compact if and only if ∥ T ∥ e = 0 ; and hence, we are going to obtain a result that implies, in many cases, the following theorem due to Ramos-Fernández and Salas-Brown [10]:

Also, it is convenient to recall that a sequence ( x n ) ⊂ X is a Schauder basis for the Banach space ( X , ‖ ⋅ ‖ X ) if for every x ∈ X we can find a complex sequence a = { a ( k ) } such that

this finally means that

With these notations, we have the following result:

The proof of Theorem 2 is divided into four steps:

Step 1. Establish limsup n → ∞ ∣ u ( n ) ∣ is an upper bound for ∥ M u ∥ e .

For any N ∈ N , we consider the set A N = { 1 , 2 , … , N } and we define the sequence u N = u ⋅ 1 A N = { u ( k ) 1 A N ( k ) } , then u N ∈ l ∞ and the operator M u N : X → X is continuous. Furthermore,

and by Theorem 1, the operator M u N : X → X is also compact. Thus, by definition of the essential norm, we can write

Next, for any sequence a = { a ( k ) } ∈ X such that ∥ a ∥ X = 1 and for any k ∈ N , we have

where

Hence, the sequence ( u − u N ) ⋅ a = { ( u ( k ) − u N ( k ) ) ⋅ a ( k ) } belongs to X and

Therefore,

and we have ∥ M u ∥ e ≤ S N for any N ∈ N , which proves that

This give us the upper bound for ‖ M u ‖ e .

Step 2. Establish that X ′ is contained into a weighted c 0 space.

The following property is a generalization of Proposition 1 in [17].

Step 3. Establish that the normalized canonical basis converges weakly to zero.

Next we shall go to prove the following result:

Step 4. Establish that limsup n → ∞ ∣ u ( n ) ∣ is a lower bound for ‖ M u ‖ e .

Now we can finish the proof of the main result. Since ∥ f n ∥ X = 1 for all n ∈ N , then by definition of the operator norm and for any compact operator K : X → X , we can write

Therefore, by taking limit when n goes to infinite, we obtain

where we have used relation (5). The result follows since the compact operator K : X → X was arbitrary.

By combining Step 1 with this last step, it follows

which completes the proof of Theorem 2.

As an immediate consequence of our result, we have (see [10,13–18]).

Final comment. There is an important result due to Axler et al. [27] about the essential norm of operators from a Banach space X into c 0 . However, there are a lot of Köthe sequence spaces X , which are not subspace of c 0 ; for instance, if ω = { ω ( n ) } is a sequence weight such that ω ( n ) > 0 for all n ∈ N , then for p > 1 , the weighted l p space, denoted by l p ( ω ) , of all complex sequences x = { x ( n ) } such that

in general, is not contained in c 0 . In these kinds of spaces, our Theorem 2 can be applied. Furthermore, our findings are applicable to various Banach sequence spaces, including c 0 , l p spaces with 1 < p < ∞ , Orlicz and Lorentz spaces, and Cesàro spaces among others, and it has as consequence the compactness results in [10,13–18]. Our result can not be applied to any Banach sequence space; for instance, if we put X = l 1 , then X ′ = X ∗ = l ∞ , and the canonical basis ( e n ) is not a Schauder basis for l ∞ because this space is not separable.