Operator ideals and three-space properties of asymptotic ideal seminorms

Abstract

We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the 'automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types p and q has Szlenk power type max{p, q}.2010 Mathematics Subject Classification. Primary 46B06, 46B20, 46B28; Secondary 46B09. Key words and phrases. Szlenk index, Szlenk power type, three-space property, operator ideals. Research of the second-named author was supported by the GAČR project 16-34860L and RVO: 67985840.

Asymptotic smoothness in Banach spaces, three space properties and applications

“…The fact that P p is a 3SP was shown in Theorem 7.5 of [10]. The proof there established an inequality similar to Theorem 5.3(i), but using a p rather than n p .…”

“…Past results. It was shown by Draga, Kochanek, and the first named author in [10] that membership in P p is a 3SP, although it was not stated in this way. We isolate here a shorter and more direct argument.…”

“…A property (P ) of Banach spaces is a Three Space Property (3SP in short) if it passes to quotients and subspaces and a Banach space X has (P ) whenever it admits a subspace Y such that Y and X/Y have (P ). The properties T p , A p , N p and P p pass quite simply to subspaces, quotients or isomorphs and it was proved in [10] that P p is a 3SP. First, we take the opportunity of this paper to provide a more direct argument for this.…”

Hamming graphs and concentration properties in non-quasi-reflexive Banach spaces

“…We recall that a three-space property is a property (P) satisfying the following: if M is a closed subspace of a Banach space X, and if two of the spaces X, M , X{M have property (P), then so has the third one. For the same reasons, being in T p , being in A p and being in N p are not three-space properties, contrary to being in P p ( [12]). Let us mention that the lack of property HC p for the space Z p can be proved directly as follows.…”

Concerning $q$-summable Szlenk index

“…subtype and subcotype of operators, defined by whether or not the operator exhibits the worst possible behavior with respect to a sequence of norms or seminorms. This practice has been undertaken by Beauzamy [1] for Radmacher subtype and subcotype to characterize when ℓ 1 or c 0 is crudely finitely representable in an operator; by Hinrichs [13] for gaussian subcotype to characterize when c 0 is crudely finitely representable in an operator; by Wenzel [21] for martingale and Haar subtype and subcotype to characterize super weak compactness; and Draga, Kochanek, and the author [8] to characterize when an operator is asymptotically uniformly smoothable, when ℓ 1 is asymptotically crudely finitely representable in an operator, and when c 0 is asymptotically crudely finitely representable in an operator.…”