New criteria of some bounded approximation properties
For Banach spaces \(X\) and \(Y\), let the Banach space of all bounded linear operators from \(X\) to \(Y\) be denoted by \({\mathcal L}(X,Y)\). Let \({\mathcal F}(X,Y)\) and \({\mathcal K}(X,Y)\) denote its subspaces of finite-rank and compact operators. A Banach space \(X\) is said to have the approximation property (AP) if the identity map \(I_X\) on \(X\) belongs to the closure of \({\mathcal F}(X,X)\) in the topology \(\tau\) of uniform convergence on compact subsets of \(X\). If \(I_X\) belongs to the \(\tau\)-closure of the subset of \({\mathcal F}(X,X)\) of the operators of norm \( \leq \lambda\), then \(X\) is said to have the \(\lambda\)-BAP. If \({\mathcal F}(X,X)\) is replaced by \({\mathcal K}(X,X)\), then the corresponding approximation properties are called the compact AP and the the compact \(\lambda\)-BAP, respectively. These are well-studied classical approximation properties. Now, if \({\mathcal F}(X,X)\) is replaced by an arbitrary linear subspace \({\mathcal A}(X,X)\) of \({\mathcal L}(X,X)\), then one obtains general versions of approximation properties, which have been studied since the 1980s by O. I. Reinov, N. Grønbæk, G. A. Willis, H.-O. Tylli, and others, including the reviewer. The author rediscovers these approximation properties (unfortunately, this has gone unnoticed by the referee) and names them the \({\mathcal A}\)-AP and the \(\lambda\)-\({\mathcal A}\)-BAP, respectively. From the main results under review, only Theorems 1.1 and 1.2 are essentially new. They state that \(X\) has the \(\lambda\)-\({\mathcal A}\)-BAP (respectively, the \({\mathcal A}\)-AP) if and only if, for every (separable reflexive) Banach space \(Y\), every \(T \in {\mathcal K}(X,Y)\) belongs to the \(\tau\)-closure of \(\{TS : S \in \mathcal A(X,X), \;\|S\| \leq \lambda\}\) (respectively, of \(\{TS : S \in \mathcal A(X,X)\}\)). The other results are known. For instance, Theorem 1.7 of the paper is contained in Corollary 1.4 of [\textit{O. I. Rejnov}, Math. Notes 33, 427--434 (1983); translation from Mat. Zametki 33, No. 6, 833--846 (1983; Zbl 0543.46009)] and Theorem 1.8 of the paper is essentially the same as Lemma 1.2 of [loc. cit.], modulo a standard perturbation argument; Theorems 1.3 and 1.4 of the paper are immediate from [\textit{A. Lissitsin, K. Mikkor} and \textit{E. Oja}, Ill. J. Math. 52, No. 2, 563--582 (2008; Zbl 1185.46009)].
scientific article; zbMATH DE number 5989926