Weak compactness of operators acting on o–O type spaces

Abstract

Abstract. We consider operators T : M 0 → Z and T : M → Z, where Z is a Banach space and (M 0 , M ) is a pair of Banach spaces belonging to a general construction in which M is defined by a "big-O" condition and M 0 is given by the corresponding "little-o" condition. Prototype examples of such spaces M are given by ℓ ∞ , weighted spaces of functions or their derivatives, bounded mean oscillation, Lipschitz-Hölder spaces, and many others. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M , weaker than the M -norm, and shows that weakly compact operators T : M 0 → Z are already quite close to being completely continuous. Further, we develop a method to extract c 0 -subsequences from sequences in M 0 . Applications are given to the characterizations of the weakly compact composition and Volterra-type integral operators on weighted spaces of analytic functions, BM OA, V M OA, and the Bloch space.

Semigroups of composition operators and Integral operators in BMOA-type spaces

“…To continue the study of compactness and weak compactness of Volterra operators on BMOA p we will need some results of K.M. Perfekt from [22], [23]. In these articles there is a general construction of pairs of spaces (M 0 , M) obtained by little oh and big oh conditions respectively.…”

“…with ϕ a the Möbius automorphisms of D. We can then apply the above theorem as in [23,Example 4] to obtain information about the Volterra operators T g acting on the spaces, thus generalizing results from [31], [9], [16], concerning compactness and weak compactness of T g on the pair (V MOA, BMOA). Theorem 7.…”

Preduals and double preduals of some Banach spaces

“…is the "little o "to BM O(Q 0 ) (for the theory of o-O spaces see [22], [23], [6]). Using the fact that cl…”

Duality and distance formulas in Lipschitz-Hölder spaces

“…The o-O type structure introduced in [23] is typical of several non-reflexive Banach spaces such as BMO and V MO (as done in the aforementioned paper), the Brezis-Bourgain-Mironescu space B (introduced in [6]) and its subspace B 0 (as done in [10], see also [11], [17]) and a particular class of Orlicz spaces (as done in [3]). This structure provides a general setting in which specific properties of Banach spaces can be shown, such as M-ideality of some subspaces (see [25]) and a characterization of weak compact operators (see [24]). Concerning lip α and Lip α 0 < α < 1, in [23] it is show that on compact subsets of R n they exhibit a o-O structure, also obtaining as a consequence the M-ideality of lip α in Lip α , which is a generalization of a result given in [5] for K = [0, 1].…”