Properties of Banach spaces, stable and unstable relative to the gap (Q1096870)

Let X, Y be closed subspaces of a Banach space Z. The opening \(\theta\) (x,y) of X and Y is defined as \[ \theta (X,Y)=\max (\sup dist_{x\in X,\quad \| x\| =1}(x,y),\quad \sup dist_{y\in Y,\quad \| y\| =1}(y,X)) \] A property P is said to be stable with respect to opening if there is a constant \(\alpha (P)>0\) such that if X has P and \(\theta (X,Y,)<\alpha (P)\), then Y has P. It was proved by \textit{M. Kadets} [Funct. Anal. Appl. 9, 156-175 (1975; Zbl 0325.46020)] that isomorphism it not stable with respect to opening. In this paper the author shows that many other properties, like the approximation property, weak sequential completeness, weak Banach Saks property, Dunford-Pettis property, C-convexity and others are not stable with respect to opening. On the other hand the author proves that quasireflexivity is stable with respect to opening.