Abstract Cesàro Spaces. I. Duality

Abstract

We study abstract Cesàro spaces CX, which may be regarded as generalizations of Cesàro sequence spaces ces p and Cesàro function spaces Ces p (I) on I = [0, 1] or I = [0, ∞), and also as the description of optimal domain from which Cesàro operator acts to X. We find the dual of such spaces in a very general situation. What is however even more important, we do it in the simplest possible way. Our proofs are more elementary than the known ones for ces p and Ces p (I). This is the point how our paper should be seen, i.e. not as generalization of known results, but rather like grasping and exhibiting the general nature of the problem, which is not so easy visible in the previous publications. Our results show also an interesting phenomenon that there is a big difference between duality in the cases of finite and infinite interval. *This publication has been produced during scholarship period of the first author at the Luleå University of Technology, thanks to a Swedish Institute scholarschip (number 0095/2013).

Abstract Cesàro Spaces. Optimal Range

“…Remark 1. If X is a symmetric space, then evidently X = X − and we get Lemma 10 from [LM14], which proof was a generalization of the Astashkin -Maligranda result from [AM09]. Moreover, our Theorem 2 includes Theorem 9 in [LM14] for the weighted L p (x α ) spaces when 1 ≤ p < ∞ and −1/p < α < 1 − 1/p.…”

“…Of course, boundedness of M on X implies also boundedness of C on X, therefore support of CX is for sure the same as support of X (cf. [LM14]). Let I = [0, ∞).…”

“…Even more can be said when the dilation operator σ a is bounded on X for a certain 0 < a < 1. Then CX is the optimal domain of C even for CX since, by Lemma 6 in [LM14], it follows that CCX = CX. On the other hand, we will see that X is the optimal range of C for X, which by the above diagram means that also for X and for CX.…”

“…For a Banach ideal space X on I = [0, 1] or I = [0, ∞) let us consider, as in [LM14], the abstract Cesàro space CX on I defined as CX = {f ∈ L 0 (I) : C|f | ∈ X} with the norm given by f…”

“…We refer the reader to [LM14], where basic facts about the spaces CX and X were presented with more details. For more references on Banach ideal spaces and symmetric spaces we refer to [KPS82], [LT79], [BS88], [KA77] and [Ma89].…”