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

Authors: Julio C. Ramos-Fernández; María A. Rivera-Sarmiento; Margot Salas-Brown
Journal: Journal of Function Spaces
 (2019)

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## 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 operatorMu acting on Cesàro sequence spaces cesp.


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## Body

## 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 spacesX and Y, whose elements are sequences of real or complex numbers, a multiplier for X and Y is defined to be a numeric sequence (real or complex) u=u(n) such that u·f∈Y for all f∈X, where · denotes pointwise function multiplication and is dropped henceforward, with no confusion resulting; that is, uf is the sequence u(n)f(n). Each multiplier u=u(n) induces a linear operator Mu:X→Y given by Mu(f)=u·f. In case Mu is continuous, we call itthe multiplication operator fromXtoYwith symbolu.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 L2(μ) of square integrable measurable functions with respect to a given measure μ.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 L2(μ). Additionally, we note that the work of Arora et al. in [4–6] examined properties of Mu 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 Lp 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 Mu acting on Lorentz sequence spaces is compact if and only if u∈c0, that is, if and only if u(n)→0 as n→∞. This last condition also characterizes the compactness of this operator Mu, 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 u between two Köthe spaces X and Y.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 operatorsMu 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 p>1 and let u=u(n) be a bounded sequence. Then for the operator Mu:cesp→cesp, we have (1)Mue=lim⁡supn→∞⁡un.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 cesp of all real sequences x=x(n) such that(2)xcesp≔∑n=1∞1n∑k=1nxkp1/p<∞, where p is a parameter fixed greater or equal than 1; in fact, Leibowitz [18] and Jagers [19] proved that ces1=0, cesp are separable reflexive Banach spaces for 1<p<∞. The space cesp arises in a natural way from Hardy’s inequality,(3)∑n=1∞1n∑k=1nxkp≤pp-1p∑k=1∞xkp which establish that the lp spaces are continuously and strictly embedded into cesp for 1<p<∞.Fork∈N fixed, the sequence ek=ek(n) defined by(4)ekn=1,n=k,0,otherwise is an element of cesp and, in this case, we have(5)ekcespp=∑n=k∞1np. Observe that ekcesp→0 as k→∞. Furthermore, the sequence of canonical vectors ekk=1∞ is an unconditional, boundedly complete, and shrinking Schauder basis for cesp (see [20]). Also, it is important to remark the following property which tell us that cesp is a solid space. More precisely, if x=x(n) and y=y(n) are real sequences such that y∈cesp with 1<p<∞ and xn≤yn for all n∈N, then x∈cesp and xcesp≤ycesp.The problem of the dual space ofcesp was solved by Jagers in [19]. He showed that cesp∗ can be identified with its Köthe dual of all the sequences y=y(n) such that(6)∑n=1∞ynxn<∞ for all real sequence x=x(n)∈cesp. In particular, this means that for each bounded linear functional F∈cesp∗ we can find a real sequence y∈c0 such that(7)Fx=∑n=1∞ynxn for all x∈cesp. We refer this fact as the representation Riesz’s theorem for the Cesàso spaces. The description of the dual of cesp given by Jagers in [19] can be quite complicated; it is given in terms of the least decreasing majorant; namely,(8)cesp∗=x=xn:xcesp∗≔∑n=1∞nx~n-x~n+1q1/q<∞,where 1/p+1/q=1 and x~ is the least decreasing majorant of x, with the property that, for n>m,(9)x~m-x~n∑k=mn-11kp-1 increases with n if m is fixed. However, Bennett [21] gave a simpler, though just isomorphic, identification of the dual space cesp∗. For 1<p<∞, Bennet considers the Banach space d(p) of all real sequence x=x(n) such that(10)xdp≔∑n=1∞supk≥nxkp1/p<∞. Observe that if x∈d(p), then x(n)→0 as n→∞ which implies that d(p) is contained in c0. Then for 1<p<∞, the dual space cesp∗ of cesp can be identified with the sequence space d(q), where 1/p+1/q=1. Furthermore, for all x∈cesp∗ Bennett shows that(11)1qxdq≤xcesp∗≤p-11/pxdq. We finish this section establishing the following property of the sequences in the space d(q) which will be of great utility for the proof of the main result of this article. A similar result for sequences in cesp appears in [20].Proposition 1.
Suppose that1<q<∞. If x=x(n)∈d(q) then(12)limn→∞⁡n1/qxn=0.Proof.
It is enough to see thatdn=nx(n)q→0 as n→∞. For any n∈N and m∈N such that m≥n we have (13)∑k=n∞supj≥k⁡xjq≥∑k=nmsupj≥k⁡xjq≥m-n+1xmq. In particular, for m=2n, we have(14)d2n=2nx2nq≤2∑k=n∞supj≥k⁡xjq+2x2nq→0 as n→∞ since x∈d(q) and x(n)→0 as n→∞. While, for m=2n-1, we have(15)d2n-1=2n-1x2n-1q≤2∑k=n∞supj≥k⁡xjq-x2n-1q→0 as n→∞. This shows the result.Remark 2.
The set of all sequencesx=x(n) satisfying the condition (12) is a Banach space with the norm(16)x=supn∈N⁡n1/qxn. More precisely, if β=β(n) is a sequence of positive numbers, the set lβ∞ of all sequences x=x(n) such that(17)xβ=supn∈N⁡βnxn<∞ is a Banach space which has a closed subspace c0β of all sequences x=x(n) such that(18)limn→∞⁡βnxn=0. This kind of spaces can be calledgrowth spaces of sequences and they are generalizations of the classical spaces l∞ and c0 (obtained when β(n)=1 for all n∈N). Hence, d(q) is contained in the growth space c0β with β(n)=n1/q for all n∈N.


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

We recall that ifX,·X is a Banach space and T:X→X is a continuous operator, then the measure of noncompactness (or essential norm) of T, denoted by Te, is the distance of T to the class of the compact operators on X; that is,(19)Te=inf⁡T-K:K:X→Xis  a  linear  compact  operator, where T denotes the operator norm of T, which is defined by T=supTfX:fX=1. It is known that an operator T:X→X is compact if and only if Te=0.In the case of multiplication operatorsMu acting on Cesàro sequence spaces, recently Komal et al. [11] (see also [22]) showed the following result.Theorem 3 (see [11]).
Suppose thatp>1 and that u=u(n) is a bounded sequence. The multiplication operator Mu:cesp→cesp is compact if and only if u∈c0 (u(n)→0 as n→∞).The condition thatu is a bounded sequence in the above theorem is necessary and sufficient for the fact that the operator Mu:cesp→cesp will be continuous or bounded. Our goal is to obtain an expression of the essential norm of Mu:cesp→cesp which implies the above result; more precisely, in our research, the result we have found is the following.Theorem 4.
Suppose thatp>1 and let u=u(n) be a bounded sequence. Then for the multiplication operator Mu:cesp→cesp we have(20)Mue=lim⁡supn→∞⁡un.Proof.
For eachN∈N, we set the sequence uN=u(1),u(2),…,u(N),0,0,…. Then by Theorem 3, the multiplication operator MuN:cesp→cesp is compact. Hence(21)Mue≤Mu-MuN=Mu-uN. But if a=a(n)∈cesp is such that acesp=1, then clearly(22)un-uNn·an≤SNan for all n∈N, where SN=sup{uk:k≥N}. Thus, since the space cesp is solid, we conclude that the sequence (u-uN)·a belongs to cesp and (23)Mu-MuN≤Mua-MuNacesp=u-uNacesp≤SNacesp=SN. Therefore Mue≤SN for all N∈N and hence(24)Mue≤lim⁡supn→∞⁡un.On the other hand, if K:cesp→cesp is any compact operator, then, for k∈N fixed, we consider the sequence ek given by(25)ekn=1,n=k,0,otherwise. We have(26)ekcespp=∑n=k∞1np≥∫k∞dxxp=1p-1k1-p and hence ek∈cesp for all k∈N and ekcesp→0 as k→∞ since it is the tail of a convergent series. Furthermore, the sequence fk⊆cesp defined by fk=ek/ekcesp satisfies that fkcesp=1 for all k∈N, and, in particular, fk is a bounded sequence in cesp. We are going to show that(27)limk→∞Kfkcesp=0.Indeed, if (27) is false, then by passing to a subsequence if is necessary and without loss of generality, we can suppose that there exists δ>0 such that(28)Kfkcesp≥δfor all k∈N. Since K:cesp→cesp is a compact operator and fk is bounded in cesp, then by passing to a subsequence if is necessary, we can suppose that K(fk) converges in cesp. That is, there exists a b∈cesp such that(29)limk→∞⁡Kfk-bcesp=0.Note that, if b=0, then this fact leads us to contradict (28). By Hahn-Banach’s theorem, it is enough to show that h(b)=0 for all bounded linear functionals h on cesp. Let h be any bounded linear functional on cesp. Then the composition hK is also a bounded linear functional on cesp, so that, by the Riesz representation theorem for cesp spaces, there exist a=a(n)∈c0 such that(30)hKc=∑n=1∞an·cn for all c=c(n)∈cesp. In particular, evaluating at fk, we have (31)hKfk=akekcesp≤k1/qak→0 as m→∞ in virtue of Proposition 1. Hence (32)hb≤hb-hKfk+hKfk≤hb-Kfk+akekcesp→0 as m→∞ and h(b)=0. This proves the claim. Next we can conclude the proof of our result. Since fk is a unitary vector in cesp for each k∈N, we can write (33)Mu-K≥Mufk-Kfkcesp≥1ekMuekcesp-Kfkcesp=1eku·ekcesp-Kfkcesp=uk-Kfkcesp. Thus taking the limit when k→∞, we conclude that(34)Mu-K≥lim⁡supk→∞uk, and therefore Mue≥lim⁡supn→∞un since the compact operator K on cesp 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 X is said to be a Köthe sequence space if it satisfies the following:(i)
Ifa,b are sequences such that a(k)≤b(k) for all k∈N and b∈X, then a∈X and aX≤bX.(ii)
IfA=n1,n2,…,nm is a finite subset of N, then the sequence 1A, defined by(35)1Ak=1,k∈A,0,k∉A,belongs toX.
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 X.



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