Approximation properties AP\(_s\) and \(p\)-nuclear operators (the case \(0<s \leq 1\))
Let \(X\) and \(Y\) be Banach spaces and let \(p>0\). Denote by \(N_p (X,Y)\) the space of all \(p\)-nuclear operators from \(X\) to \(Y\) and by \(N_p^{\text{reg}}(X,Y)\) the space of all bounded linear operators \(T\) from \(X\) to \(Y\) that are contained in \(N_p (X,Y^{\ast\ast})\) whenever \(T\) is viewed as an operator from \(X\) to \(Y^{\ast\ast}\). By \textit{A. Grothendieck} [Mem. Am. Math. Soc. 16 (1955; Zbl 0123.30301)] and \textit{E. Oja} and \textit{O. Reinov} [C. R. Acad. Sci., Paris, Sér. I, 305, 121--122 (1987; Zbl 0621.46016); Izv. Akad. Nauk Est. SSR, Fiz., Mat. 37, 14--17 (1988; Zbl 0669.46008)], if \(X^\ast\) or \(Y^{\ast\ast\ast}\) has the approximation property (AP for short), then \(N_1^{\text{reg}}(X,Y)\subset N_1(X,Y)\); there exists a Banach space \(Z\) such that \(Z^{\ast\ast}\) has a basis (while \(Z^{\ast\ast\ast}\) does not have the \(AP\)) and \(N_1^{\text{reg}}(Z^{\ast\ast},Z)\not\subset N_1(Z^{\ast\ast},Z)\). These results are extended and refined for the case of Reinov's approximation properties AP\(_s\) of order \(s\), \(0<s\leq 1\), (recall that AP\(_1\) is the usual AP) and their generalized version, introduced in this article. Among others, the following results are proved. If \(0<s\leq 1\) and \(X^\ast\) or \(Y^{\ast\ast\ast}\) has the AP\(_s\), then \(N_s^{\text{reg}}(X,Y)\subset N_1(X,Y)\). If \(2/3\leq s<1\), then there exists a Banach space \(Z\) such that \(Z^{\ast\ast}\) has a basis, all spaces \(Z\), \(Z^\ast\), \(Z^{\ast\ast}\), \dots{} have the AP\(_s\), \(Z^{\ast\ast\ast}\) does not have the \(AP_r\) and \(N_r^{\text{reg}}(Z^{\ast\ast},Z)\not\subset N_1(Z^{\ast\ast},Z)\) whenever \(s<r\leq 1\). If \(2/3\leq s<1\), then there exist Banach spaces \(W\) and \(Z\) such that \(W\) and \(Z^{\ast\ast}\) have bases and \(N_r^{\text{reg}}(W,Z)\not\subset N_p(W,Z)\) whenever \(s<r\leq 1\) and \(1/p=1/s-1/2\) (note that \(1\leq p<2\)). The latter result as well as some others in this article improve previous results of the author [Teor. Oper. Teor. Funkts. 1, 145--165 (1983; Zbl 0561.46033)].
scientific article; zbMATH DE number 1971373
scientific article