The dual form of the approximation property for a Banach space and a subspace

T. Figiel¹ ; W. B. Johnson²

¹ Institute of Mathematics Polish Academy of Sciences Branch in Gdańsk Wita Stwosza 57 80-952 Gdańsk, Poland
² Department of Mathematics Texas A&M University College Station, TX 77843-3368, U.S.A.

Studia Mathematica, Tome 231 (2015) no. 3, pp. 287-292

Résumé

Given a Banach space $X$ and a subspace $Y$, the pair $(X,Y)$ is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on $X$ all of which leave the subspace $Y$ invariant such that the net converges uniformly on compact subsets of $X$ to the identity operator. In particular, if the pair $(X,Y)$ has the AP then $X$, $Y$, and the quotient space $X/Y$ have the classical Grothendieck AP. The main result is an easy to apply dual formulation of this property. Applications are given to three-space properties; in particular, if $X$ has the approximation property and its subspace $Y$ is ${\mathcal {L}}_\infty $, then $X/Y$ has the approximation property.

Détail

DOI : 10.4064/sm8367-2-2016

Keywords: given banach space subspace pair said have approximation property provided there net finite rank bounded linear operators which leave subspace invariant net converges uniformly compact subsets identity operator particular pair has quotient space have classical grothendieck main result easy apply dual formulation property applications given three space properties particular has approximation property its subspace mathcal infty has approximation property