On the Essential Norm of Multiplications Operators Acting on Cesàro Sequence Spaces

Abstract

In this article, we establish an important property about the growth of sequences in the dual space of Cesàro sequence spaces. As a consequence of this fact, we calculate the measure of noncompactness or the essential norm of the multiplication operator acting on Cesàro sequence spaces .

1. Introduction

During the past decade, there has been increasing interest in properties of multipliers between functional Banach spaces of real or complex sequences. Given two Banach spaces and , whose elements are sequences of real or complex numbers, a multiplier for and is defined to be a numeric sequence (real or complex) such that for all , where denotes pointwise function multiplication and is dropped henceforward, with no confusion resulting; that is, is the sequence . Each multiplier induces a linear operator given by . In case is continuous, we call it the multiplication operator from to with symbol .

Multiplication operators are classical and continue to be widely studied. In particular, multipliers of spaces of measurable functions were thoroughly examined during the mid-twentieth century. For example, Halmos’s monograph [1, Ch. 7] contains important information about multiplication operators on the Hilbert space of square integrable measurable functions with respect to a given measure .

We also mention here the pioneering work Singh/Kumar in [2, 3] on properties of multiplication operators acting on spaces of measurable functions. These authors studied the compactness and closedness of the range of multiplication operators on . Additionally, we note that the work of Arora et al. in [4–6] examined properties of on Lorentz and Lorentz-Bochner spaces. Further significant results regarding multiplication operators were obtained by Castillo et al. in [7–9], in which these authors showed that the techniques used by previously mentioned authors can be modified to study multiplication operators on weak spaces, Orlicz-Lorentz spaces, and variable Lebesgue spaces.

Motivated by the idea of finding nonnull compact multiplication operators, recently, interesting works about properties of multiplication operators acting on Banach sequences spaces have appeared. Arora et al. in [6] show that a multiplication operator acting on Lorentz sequence spaces is compact if and only if , that is, if and only if as . This last condition also characterizes the compactness of this operator , acting on other Banach sequence spaces such as Orlicz-Lorentz sequence spaces [10], Cesàro sequence spaces [11], Cesàro-Orlicz sequence spaces [12], among others. However, the above spaces are classified as Köthe sequence spaces and the characterization of compactness (and other properties) of multiplication operators acting on Köthe sequence spaces is due to Ramos-Fernández and Salas-Brown [13] (see also [14, 15]). It is an open problem to characterize the multipliers between two Köthe spaces and .

Related to the problem of the characterization of the compactness of multiplication operators we have the problem of estimating its essential norm or the noncompactness measure of multiplication operators. This subject has been widely studied in the context of analytic functions. In the case of multiplication operators acting on nonatomic Köthe spaces the essential norm was calculate by Castillo et al. [15] and it is an open problem calculates the essential norm of this operator in a more general sense. The essential norm of multiplication operators acting on Lorentz sequence spaces was calculated by Castillo et al. [16]. The main objective of this article is to calculate the essential norm of multiplication operators acting on Cesàro sequence spaces. More precisely, we are going to show the following result.

Main Theorem. Suppose that and let be a bounded sequence. Then for the operator , we have We present the proof of the above result in Section 3, and in Section 2 we gather some properties of the Cesàro sequence spaces and its dual given a new property for sequences in this space.

2. Some Remarks of Cesàro Sequence Spaces

The Cesàro sequence spaces appeared in 1968 in connection with the problem of the Dutch Mathematical Society to find their duals. This sequence space was first defined by Shiue [17] in 1970 as the set of all real sequences such that where is a parameter fixed greater or equal than ; in fact, Leibowitz [18] and Jagers [19] proved that , are separable reflexive Banach spaces for . The space arises in a natural way from Hardy’s inequality, which establish that the spaces are continuously and strictly embedded into for .

For fixed, the sequence defined by is an element of and, in this case, we have Observe that as . Furthermore, the sequence of canonical vectors is an unconditional, boundedly complete, and shrinking Schauder basis for (see [20]). Also, it is important to remark the following property which tell us that is a solid space. More precisely, if and are real sequences such that with and for all , then and .

The problem of the dual space of was solved by Jagers in [19]. He showed that can be identified with its Köthe dual of all the sequences such that for all real sequence . In particular, this means that for each bounded linear functional we can find a real sequence such that for all . We refer this fact as the representation Riesz’s theorem for the Cesàso spaces. The description of the dual of given by Jagers in [19] can be quite complicated; it is given in terms of the least decreasing majorant; namely,where and is the least decreasing majorant of , with the property that, for , increases with if is fixed. However, Bennett [21] gave a simpler, though just isomorphic, identification of the dual space . For , Bennet considers the Banach space of all real sequence such that Observe that if , then as which implies that is contained in . Then for , the dual space of can be identified with the sequence space , where . Furthermore, for all Bennett shows that We finish this section establishing the following property of the sequences in the space which will be of great utility for the proof of the main result of this article. A similar result for sequences in appears in [20].

Proposition 1. Suppose that . If then

Proof. It is enough to see that as . For any and such that we have In particular, for , we have as since and as . While, for , we have as . This shows the result.

Remark 2. The set of all sequences satisfying the condition (12) is a Banach space with the norm More precisely, if is a sequence of positive numbers, the set of all sequences such that is a Banach space which has a closed subspace of all sequences such that This kind of spaces can be called growth spaces of sequences and they are generalizations of the classical spaces and (obtained when for all ). Hence, is contained in the growth space with for all .

3. The Essential Norm of Multiplication Operators on Cesàro Sequence Spaces

We recall that if is a Banach space and is a continuous operator, then the measure of noncompactness (or essential norm) of , denoted by , is the distance of to the class of the compact operators on ; that is, where denotes the operator norm of , which is defined by . It is known that an operator is compact if and only if .

In the case of multiplication operators acting on Cesàro sequence spaces, recently Komal et al. [11] (see also [22]) showed the following result.

Theorem 3 (see [11]). Suppose that and that is a bounded sequence. The multiplication operator is compact if and only if ( as ).

The condition that is a bounded sequence in the above theorem is necessary and sufficient for the fact that the operator will be continuous or bounded. Our goal is to obtain an expression of the essential norm of which implies the above result; more precisely, in our research, the result we have found is the following.

Theorem 4. Suppose that and let be a bounded sequence. Then for the multiplication operator we have

Proof. For each , we set the sequence . Then by Theorem 3, the multiplication operator is compact. Hence But if is such that , then clearly for all , where . Thus, since the space is solid, we conclude that the sequence belongs to and Therefore for all and henceOn the other hand, if is any compact operator, then, for fixed, we consider the sequence given by We have and hence for all and as since it is the tail of a convergent series. Furthermore, the sequence defined by satisfies that for all , and, in particular, is a bounded sequence in . We are going to show thatIndeed, if (27) is false, then by passing to a subsequence if is necessary and without loss of generality, we can suppose that there exists such thatfor all . Since is a compact operator and is bounded in , then by passing to a subsequence if is necessary, we can suppose that converges in . That is, there exists a such thatNote that, if , then this fact leads us to contradict (28). By Hahn-Banach’s theorem, it is enough to show that for all bounded linear functionals on . Let be any bounded linear functional on . Then the composition is also a bounded linear functional on , so that, by the Riesz representation theorem for spaces, there exist such that for all . In particular, evaluating at , we have as in virtue of Proposition 1. Hence as and . This proves the claim. Next we can conclude the proof of our result. Since is a unitary vector in for each , we can write Thus taking the limit when , we conclude that and therefore since the compact operator on was arbitrary. This finishes the proof.

Remark 5. We finish this article with the following comment: the fact that the result obtained in this article is also valid for the other Banach spaces of sequences as, for example, Lorentz sequence spaces such as that we can see in [16], is very interesting. Lorentz sequence spaces and Cesàro sequence spaces are examples of Köthe sequence spaces. A Banach space of sequences is said to be a Köthe sequence space if it satisfies the following:(i)If are sequences such that for all and , then and .(ii)If is a finite subset of , then the sequence , defined by belongs to . Thus, it is natural to ask if the result obtained in this article (and in [16]) is also valid for multiplication operator acting on any Köthe sequence space .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.