On null sequences for Banach operator ideals, trace duality and approximation properties

LASSALLE, SILVIA BEATRIZ; TURCO, PABLO

MATHEMATISCHE NACHRICHTEN

WILEY-V C H VERLAG GMBH

Lugar: Weinheim; Año: 2017 vol. 290 p. 2308 - 2321

0025-584X

Let $A$ be a Banach operator ideal and $X$ be a Banach space. We undertake the study of the vector space of $A$-null sequences of Carl and Stephani on $X$, $c_{0,A}(X)$, from a unified point of view after we introduce a norm which makes it a Banach space. To give accurate results we consider local versions of the different types of accessibility of Banach operator ideals. We show that in the most common situations, when $A$ is right-accessible for $(ell_1;X)$, $c_{0,A}(X)$ behaves much alike $c_0(X)$. When this is the case we give a geometric tensor product representation of $c_{0,A}(X)$. On the other hand, we show an example where the representation fails. Also, via a trace duality formula, we characterize the dual space of $c_{0,A}(X)$. We apply our results to study some problems related with the $K_A$-approximation property giving a trace condition which is used to solve the remaining case ($p=1$) of a problem posed by Kim (2015). Namely, we prove that if a dual space has the $K_1$-approximation property then the space has the $K_{u1}$-approximation property.