Abstract Cesàro spaces: Integral representations

Abstract

The Cesàro function spaces Ces p = [C, L p ], 1 ≤ p ≤ ∞, have received renewed attention in recent years. Many properties of [C, L p ] are known. Less is known about [C, X] when the Cesàro operator takes its values in a rearrangement invariant (r.i.) space X other than L p . In this paper we study the spaces [C, X] via the methods of vector measures and vector integration. These techniques allow us to identify the absolutely continuous part of [C, X] and the Fatou completion of [C, X]; to show that [C, X] is never reflexive and never r.i.; to identify when [C, X] is weakly sequentially complete, when it is isomorphic to an AL-space, and when it has the Dunford-Pettis property. The same techniques are used to analyze the operator C : [C, X] → X; it is never compact but, it can be completely continuous.

Local approach to order continuity in Cesàro function spaces

“…for 0 < x ∈ I. For a Banach ideal space X on I we define an abstract Cesàro space CX = CX(I) by CX := {f ∈ L 0 (I) : C|f | ∈ X}, with the norm f CX = C|f | X (see [9], [10], [22], [23], [24]). Let us note that for nonsymmetric space X the space CX need not have a weak unit even if X has it (see [22,Example 2]), so in general supp(CX) ⊂ supp(X).…”

“…Here, we specify Theorems A, B, C, D and E, Fact 1 and Lemma 2 because we will used them often in this article.Section 3 contains the main results of this paper. Curbera and Ricker in [9] proved that (CX) a = C(X a ) for symmetric spaces X = L ∞ [0, 1] on I = [0, 1]. Moreover, Kiwerski and Kolwicz in [15] have shown analogous equality in the case of the Cesàro-Orlicz function spaces Ces Φ := CL Φ , see also Remark 6 for a more accurate discussion.…”

The weak Banach–Saks property for function spaces

“…Due to the fact that the vector measure m is positive we have [21,Theorem 3.13]. This, together with the fact (due to the supports of the functions h ′ n , n ∈ N, being disjoint) that n k=1 |h ′ k | = | n k=1 h ′ k | for n ∈ N implies, from (7), that (8) lim…”

Isometric Copies of $$\varvec{l^\infty }$$ l ∞ in Cesàro–Orlicz Function Spaces

“…The structure of different types of Cesàro spaces has been widely investigated during the last decades from the isomorphic as well as isometric point of view. The spaces generated by the Cesàro operator (including abstract Cesàro spaces) have been considered by Curbera, Ricker and Leśnik, Maligranda in several papers (see [12][13][14][15][27][28][29]). The classical Cesàro sequence spaces ces p and function spaces Ces p have been studied by many authors (see [2,3]-also for further references, [1,4,10,11]).…”