Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l^∞ spaces

1 Introduction

Throughout history, the importance of using numerical matrices to solve problems in the natural sciences and their applications has been noted. In particular, tridiagonal matrices commonly appear in the applications of mathematics and physics. For example, in the discrete approximation using the finite difference method for the linear second-order differential equation, we can establish a relationship between the solution and the boundary conditions through a matrix of this type (see Burden et al. [1] and LeVeque [2]). In this case, we can imagine that an infinite matrix could represent the “ideal case” of discretization, where the error of the approximation is zero.

An infinite matrix M is a function M : N × N → C , which, as is usual, is denoted by M = ( m i , j ) , where m i , j represents the entry at row i ∈ N and column j ∈ N , where N = { 1 , 2 , 3 , … } is the set of all positive integers and C is the set of complex numbers. The multiplication of an infinite matrix M by a sequence x = ( x k ) is realized in the usual way. In fact, we have M × x = y = ( y i ) with y i = ∑ j = 1 ∞ m i , j x j whenever this series will be a convergent series for all i ∈ N . If X and Y are Banach sequence spaces and y = M × x ∈ Y for all x ∈ X , then the infinite matrix M define a linear transformation T M : X → Y given by T M ( x ) = M × x , and we are interested in studying its topological properties, such as continuity and the compactness.

The study of the infinite matrices is more or less ancient, dating back to around 1884 with the work of Poincaré, who wanted to provide a rigorous foundation for the use of infinite matrices and the calculus of determinants. The interest increased, and the study of these matrices was continued by Helge Von Koch and was further encouraged by David Hilbert, who studied the eigenvalues of integral operators by viewing these as infinite matrices. For a historical overview and recent applications of infinite matrices, we refer to the monographs by Cooke [3], Shivakumar et al. [4], and the article by Bernkopf [5].

The continuity problem of a process defined by an infinite matrix M consists of finding conditions on M such that a Banach space of sequences X will be mapped into another space Y by the application of an infinite matrix M . This problem is related to the stability of the process defined by an infinite matrix and has been studied by numerous mathematicians in different sequence spaces. In [6], Stieglitz and Tietz provide a table with the conditions required for an infinite matrix M to apply a classical sequence space (such as c , c 0 , l p , etc.) into another space.

In [6], we can see that famous mathematicians such as Hahn, Hardy, Littlewood, Lorentz, Orlicz, Schur, Toeplitz, among others have studied this problem, which is still of great interest, specially in particular cases of special matrices such as diagonal, bidiagonal, tridiagonal, triangular, or matrices defined by blocks.

An operator T : X → Y is said to be compact if T maps bounded subsets of X to relatively compact subsets of Y . Compact operators are rich in properties, and they find applications in the problem of determining the solution of a linear system of infinite equations of the form ( λ K + I ) x = y , where x ∈ X is the unknown sequence, K : X → X is a compact operator, λ is a non-zero scalar, and y ∈ X is given. In the article by González et al. [7], historical remarks about compact operators and their applications can be found. There is a growing interest in establishing or characterizing the compactness of certain operators acting on specific Banach spaces of sequences. The compactness of a diagonal operator acting on very general Köthe sequence spaces was characterized by Ramos-Fernández and Salas-Brown in [8]. More precisely, Ramos-Fernández and Salas-Brown in [8] proved that if X is a Köthe sequence space and u = ( u k ) is a bounded sequence, then the multiplication operator M u : X → X defined by M u ( x ) = u k ⋅ x k is a compact operator if and only if lim k → ∞ ∣ u k ∣ = 0 (i.e., u ∈ c 0 ). In the article by El-Shabrawy [9], we can find characterizations of the compactness of the Rhaly operator (which generalize Cesàro matrix operator) and the generalized difference operator (or bidiagonal operator) between a lot of Banach sequence spaces such as h (Hahn sequence space), c s , l 1 among others. Our intention in this article is to estimate the essential norm of tridiagonal operators acting from weighted Orlicz or weighted l ∞ sequence spaces into another of the same kind, so that we can obtain characterizations of the compactness of this operator in several cases.

An infinite matrix M is called tridiagonal if there exist numerical sequences a = ( a k ) , b = ( b k ) and c = ( c k ) such that m k , k = a k , m k − 1 , k = b k , m k + 1 , k = c k for all k ∈ N and m i , j = 0 otherwise. In this case, the tridiagonal matrix will be denoted by Δ a , b , c . The multiplication of an infinite tridiagonal matrix Δ a , b , c by a numerical sequence x = ( x k ) is the sequence y = ( y k ) defined by y 1 = a 1 x 1 + c 1 x 2 and

for all k ≥ 2 . This product is represented by

and can be seen as a simple process in which the entry x is converted in the output y by application of an infinite tridiagonal matrix Δ a , b , c . Thus, this process allows us to define a linear operator T a , b , c acting on a Banach sequence space X . In fact, if x = ( x k ) is a complex sequence and belongs to the Banach sequence space X , then we can define y = T a , b , c ( x ) = Δ a , b , c × x , and we are interested in the topological properties of these operators, such as continuity, compactness, and to estimate its essential norm.

Clearly, the operator T a , b , c is linear and generalize various types of linear transformations from several perspectives. The first one is the well-known multiplication operator, which is obtained when the sequences b and c are the null sequence 0 , that is, when b k = c k = 0 for all k ∈ N . In this case, it is usual to write M a instead of T a , 0,0 . Multiplication operators have been studied by several researchers and is still a subject of interest for many mathematicians (see, for instance, [8] and the references therein). In fact, tridiagonal operators can be seen as perturbations of multiplication operators in the following sense: If we denote by L the left shift operator and by R the right shift operator, that is,

then we have

where for any sequence y = ( y k ) , we define y 0 = 0 .

If only c is the null sequence 0 , then we have the generalized difference operator Δ a , b or bidiagonal operator which has been studied by Akhmedov and El-Shabrawy [10] and by a lot of researchers (see [9] and the references therein). If each of the sequences a , b , and c are constants sequences of complex numbers, then the infinite tridiagonal matrix is a special case of the infinite Toeplitz matrix and it is well known that the Toeplitz operator is bounded if and only if the coefficients of the Toeplitz matrix are the Fourier coefficients of some essentially bounded function f . In fact, there has been recent work on weighted Toeplitz operators acting on weighted sequence spaces [11].

The compactness and the estimation of the essential norm of the tridiagonal operator T a , b , c acting on weighted l 2 spaces were studied recently by Caicedo et al. [12]. Our goal is to extend the results obtained by Caicedo et al. [12] to the more general case of weighted Orlicz spaces, which we define in Section 2. Our study also will be an extension and a generalization of recent results due to Ramos-Fernández and Salas-Brown [13] regarding the compactness of multiplication operators acting on Orlicz sequence spaces. In addition, we estimate the essential norm of the tridiagonal operators T a , b , c in the case when a weighted Orlicz spaces is mapped into a weighted l ∞ space and vice versa. We also consider the case when T a , b , c maps a weighted l ∞ space into itself. Finally, in the last section, we extend our results to another kind of weighted Orlicz sequence space.

The structure of the article is as follows. In Section 2, we define a class of weighted Orlicz sequence spaces and present some of their properties. In Section 3, we address the problem of the continuity of tridiagonal operators, specifically when they map l φ ( ω ) into itself, when they map l ∞ ( ω ) into itself, and when they map l φ ( ω ) into l ∞ ( ω ) . In Section 4, we estimate the essential norm of the tridiagonal operators for the cases studied in Section 3: l φ ( ω ) → l φ ( ω ) , l ∞ ( ω ) → l ∞ ( ω ) , and l φ ( ω ) → l ∞ ( ω ) . Finally, in Section 5, we examine other types of weighted Orlicz sequence spaces, we characterize the conditions under which the tridiagonal operator is continuous on these spaces, and we compute its essential norm when acting on these spaces of Orlicz sequences.

2 Some remarks on weighted Orlicz sequence spaces

In this section, we present some properties of the weighted Orlicz sequence spaces. First, we recall that a function φ : [ 0 , ∞ ) → [ 0 , ∞ ) is said to be a Young function (or N -function) if it is an increasing, continuous and convex function which assume the value zero only at zero and φ ( t ) → ∞ as t → ∞ . A Young function φ satisfies the global Δ 2 -condition, denoted as φ ∈ Δ 2 , if there exists a constant K > 0 such that φ ( 2 t ) ≤ K φ ( t ) for all t ≥ 0 . A numerical sequence ω = ( ω k ) such that ω k > 0 for all k ∈ N will be called a weight. Given a Young function φ , a weight ω and a complex sequence x = ( x k ) , we set

and then we say that x belongs to the weighted Orlicz sequence space l φ ( ω ) if there exists a λ > 0 such that N ω φ x λ ≤ 1 . That is,

Note that the convexity of φ implies that l φ ( ω ) is a vector space. When φ ( t ) = 1 p t p with p > 1 fixed, we obtain the weighted l p space (see, for instance, [14] for the case p = 2 ) and when ω ≡ 1 , i.e., ω k = 1 for all k ∈ N , then we have the Orlicz sequence space l φ . This kind of weighted Orlicz sequence spaces has been considered by some researchers, for instance, in [15], the authors considered this kind of spaces for functions defined on a group. Note also that x ∈ l φ ( ω ) if and only if M ω ( x ) = x ⋅ ω ∈ l φ . This last relation implies that the multiplication operator M ω : l φ ( ω ) → l φ is a linear bijection. Furthermore, we can observe that for x ∈ l φ ( ω ) , the set

is not empty, clearly, it has lower bound 0. Observe that if λ 0 ∈ E ω φ ( x ) and λ > λ 0 , then λ ∈ E ω φ ( x ) . Thus, we can define for x ∈ l φ ( ω )

and ∥ ⋅ ∥ φ , ω is a norm for l φ ( ω ) , which is known as the Luxemburg norm of x ω . In fact, the pair ( l φ ( ω ) , ∥ ⋅ ∥ φ , ω ) is a Banach space. In the case that ω k = 1 for all k ∈ N , we write ∥ ⋅ ∥ φ instead of ∥ ⋅ ∥ φ , ω . Also, it is important to note that ∥ x ∥ φ , ω = ∥ M ω ( x ) ∥ φ and M ω : l φ ( ω ) → l φ is a surjective linear isometry. We refer to the excellent book by Rao and Ren [16] for more properties of Young functions and more properties of Orlicz spaces. Here, we include some facts that we are going to use throughout this article.

1. If x ∈ l φ ( ω ) and x is not the null sequence, then

   (5) N ω φ x ∥ x ∥ φ , ω ≤ 1 .

   That is, ∥ x ∥ φ , ω ∈ E ω φ ( x ) and ∥ x ∥ φ , ω is, in fact, a minimum.

2. For all x ∈ l φ ( ω ) , we have

   (6) ∥ x ∥ φ , ω ≤ 1 ⇔ N ω φ ( x ) ≤ 1 .

3. If x ∈ l φ ( ω ) , then lim k → ∞ ∣ x k ∣ ω k = 0 . This relation implies that l φ ( ω ) is a subspace of the weighted c 0 space, which we denote by c 0 ( ω ) . Observe that when ω k = 1 for all k ∈ N , c 0 ( ω ) coincides with the classical c 0 space.

4. The space l φ ( ω ) is continuously embedded in l ∞ ( ω ) . That is,

   (7) ∥ x ∥ ∞ , ω ≤ φ − 1 ( 1 ) ∥ x ∥ φ , ω ,

   for all x ∈ l φ ( ω ) , where ∥ x ∥ ∞ , ω = sup k ∈ N ∣ x k ∣ ω k . Here, l ∞ ( ω ) is the set of all numerical sequences x = ( x k ) such that ∥ x ∥ ∞ , ω < ∞ .

5. For n ∈ N fixed, the operator F n : l φ ( ω ) → C defined by F n ( x ) = x n (evaluation functional at n ) is bounded and

   (8) ∥ F n ∥ = 1 ∥ e n ∥ φ , ω ,

   where ‖ ⋅ ‖ is the operator norm and e n = ( e n , k ) is the n th canonical sequence defined as e n , n = 1 and e n , k = 0 for k ≠ n .

Given a Young function φ , its complementary function ψ : [ 0 , ∞ ) → [ 0 , ∞ ) is defined by

We can see that ψ is a Young function. Furthermore, the Young inequality, given by τ t ≤ φ ( τ ) + ψ ( t ) for all t , τ ≥ 0 , is immediately evident. Also it is known that if φ ∈ Δ 2 , then the dual space of l φ is precisely l ψ in the sense that for each F ∈ ( l φ ) ′ , the dual of l φ , there exists y ∈ l ψ such that F ( x ) = ∑ k = 1 ∞ x k y k , for all x ∈ l φ . Thus, using the fact that the multiplication operator M ω : l φ ( ω ) → l φ defined by M ω ( x ) = x ⋅ ω is a surjective linear isometry, we obtain ( l φ ( ω ) ) ′ = l ψ ( ω ) in the sense that for each G ∈ ( l φ ( ω ) ) ′ , there exists y ∈ l ψ ( ω ) such that

for all x ∈ l φ ( ω ) . This last affirmation can be seen with the following diagram:

We refer this last fact as Riesz’s representation theorem for l φ ( ω ) . We would like recommend the excellent book by Rao and Ren [16] for more properties of Orlicz spaces.

3 On the continuity of tridiagonal operators between l φ ( ω ) and l ∞ ( ω ) spaces

In this section, we shall characterize all sequences a , b , c , which define continuous tridiagonal operator acting between l φ ( ω ) and l ∞ ( ω ) spaces. The results presented in this section are genuine extensions and generalizations of those found in [12,13]. Herein, we use the notation A ≃ B if there exists a constant K > 0 , independent of A and B , such that K − 1 A ≤ B ≤ K A . We note that

where e n is the n th canonical sequences. Also, we can see that

that is, if T a , b , c ( e n ) = y = ( y k ) , then y n − 1 = c n − 1 , y n = a n , y n + 1 = b n and y k = 0 otherwise (recall that, for convenience, we have defined y 0 = 0 for any sequence y = ( y k ) , although our sequences always start at index 1). Hence, for each n ≥ 1 , we have

and we have the following result about the continuity of the tridiagonal operator.

We can also characterize the continuity of the tridiagonal operator from l ∞ ( ω ) to l φ ( ω ) . In the following result, we present the necessary and sufficient conditions:

We conclude this section with the following remark: