Sequential w‐right continuity and summing operators

Abstract

We continue the study of the w-right and strong * topologies in general Banach spaces started in [36,37] and [35]. We show that in L1 (μ)-spaces the w-right convergence of sequences admits a simpler control. Some considerations about these topologies will be contemplated in the particular cases of C*-algebras and JB*-triples in connection with summing operators. We also study (sequential) w-right-norm and strong*-norm continuity for holomorphic mappings. and the second author of the present note introduced in [36] two interesting topologies: the strong * and the w-right topology in the following way: let X and Y be two Banach spaces, for every bounded linear operator T : X −→ Y , we can consider a seminorm on X defined by x T := T (x) . The strong * -topology is the topology generated by the family of seminorms · T , where T : X −→ H is a bounded linear operator from X to some Hilbert space H (such a topology is denoted by S * (X, X * )). Similarly, the w-right-topology is the topology generated by the family of seminorms · T where T runs in the set of all bounded linear operators from X to a reflexive space [36].In Section 2, we establish new methods for controlling w-right convergent sequences in L 1 (μ) spaces. Section 3 is devoted to a more detailed study of strong*-norm continuous operators between Banach spaces. In the particular cases of operators whose domain is a C*-algebra or a JB*-triple, we explore the connections with p-C*-summing and p-JB*-triple-summing operators. We prove an extension property for 2-C*-summing and 2-JB*-triple-summing operators (see Theorems 3.6 and 3.9). In this section we shall also introduce and develop p-JB*-triple-summing operators on JB*-triples as suitable generalizations of p-C*-summing operators on C*-algebras in the sense introduced by Pisier in [39].The last section of the paper is devoted to the study of those holomorphic mappings of bounded type which are sequentially w-right-norm continuous. The main result in [35] establishes that a bounded linear operator T : X −→ Y is weakly compact if and only if T is w-right-norm continuous. We shall provide examples showing that none of these implications holds for continuous polynomials in general Banach spaces. In the linear case, T is weakly compact if and only if T * * is Y -valued. In the setting of multilinear operators, this equivalence has been recently studied in [37]. One of the main results in the just quoted paper proves that when X 1 , . . . , X k are non zero sequentially right Banach spaces and T : X 1 × · · · × X k −→ X is a multilinear operator, then T is RQCC (i.e., T is jointly sequentially w-right-norm continuous) if and only if all of the Aron-Berner extensions of T are X-valued if and only if T has an X-valued Aron-Berner extension. We shall consider here holomorphic mappings of bounded type f between two Banach spaces X and Y with X being a sequentially right Banach

Sharp coincidences for absolutely summing multilinear operators

“…where E * denotes the topological dual of E and B E * represents its closed unit ball (for the theory of absolutely summing operators we refer to [20] and, for recent results, [14,21] and references therein). The space of all absolutely (q; p)-summing operators from E to F is denoted by Π q;p (E; F ) (or Π p (E; F ) if p = q).…”

“…where E * denotes the topological dual of E and B E * represents its closed unit ball (for the theory of absolutely summing operators we refer to [20] and, for recent results, [14,21] and references therein).…”

Abstract extrapolation theorems for absolutely summing nonlinear operators

“…The space of all absolutely p-summing linear operators from X to Y is represented by Π p (X; Y ). We refer the interested reader to [9,15,16] for more recent results and further details. A famous result due to Grothendieck asserts that every continuous linear operator from 1 to 2 is absolutely p-summing, regardless of the p ≥ 1.…”

A general Extrapolation Theorem for absolutely summing operators

“…A continuous linear operator u : X → Y is absolutely p-summing (we write u ∈ Π p (X; Y )) if (u(x j )) ∞ j=1 ∈ ℓ p (Y ) whenever (x j ) ∞ j=1 ∈ ℓ p,w (X) . For details we refer to the classical monograph [8] and to [4,9] for more recent results.…”