Some duality results on bounded approximation properties of pairs

Eve Oja ¹   ; Silja Treialt ²

¹ Faculty of Mathematics and Computer Science Tartu University J. Liivi 2 50409 Tartu, Estonia and Estonian Academy of Sciences Kohtu 6 10130 Tallinn, Estonia
² Faculty of Mathematics and Computer Science Tartu University J. Liivi 2 50409 Tartu, Estonia

Studia Mathematica, Tome 217 (2013) no. 1, pp. 79-94

Résumé

The main result is as follows. Let $X$ be a Banach space and let $Y$ be a closed subspace of $X$. Assume that the pair $(X^{*}, Y^{\perp })$ has the $\lambda $-bounded approximation property. Then there exists a net $( S_\alpha )$ of finite-rank operators on $X$ such that $S_\alpha (Y) \subset Y$ and $\| S_\alpha \| \leq \lambda $ for all $\alpha $, and $( S_\alpha )$ and $( S^{*}_\alpha )$ converge pointwise to the identity operators on $X$ and $X^{*}$, respectively. This means that the pair $(X,Y)$ has the $\lambda $-bounded duality approximation property.

Détail

DOI : 10.4064/sm217-1-5

Keywords: main result follows banach space closed subspace nbsp assume pair * perp has lambda bounded approximation property there exists net alpha finite rank operators alpha subset alpha leq lambda alpha alpha * alpha converge pointwise identity operators * respectively means pair has lambda bounded duality approximation property