On the best compact approximation problem for operators between \(L_ p\)- spaces
A pair of Banach spaces (X,Y) is said to have the best compact approximation property (b.c.a.p.) if, for every bounded linear operator \(T: X\to Y\) there exists a compact operator \(K: X\to Y\) satisfying \(\| T-K\| \leq \| T-K'\|\) for all compact K'. We say that X has the b.c.a.p. if the pair (X,X) has this property. It is known that \(\ell_ p\) has the b.c.a.p. for \(1\leq p<\infty\) while \(\ell_{\infty}\), \(L_ 1\), \(L_{\infty}\) and C[0,1] fail to have it. \textit{V. Benyamini} and \textit{P. K. Lin} [Israel J. Math. 51, 298-304 (1985; Zbl 0596.47020)] showed that for \(1<p<\infty\), \(p\neq 2\), Lp fails the best compact approximation property. In this paper this result is extended by showing that \((\ell_ p,L_ p)\) fails the b.c.a.p. for \(1<p<2\). A sufficient condition is also given for an operator \(T: L_ p\to L_ p\) \((2<p<\infty)\) to have a best compact approximant (b.c.a.). Main results: 1. Let \(1<p<2\). Then \((\ell_ p,L_ p)\) fails the b.c.a.p. 2. Let \(2<p<p\) and let \(T: L_ p\to L_ p\) be a bounded linear operator satisfying: If \((x_ n)\) is a uniformly bounded weakly null sequence in \(L_ p\), then \(\| Tx_ n\| \to 0\) then T has a best compact approximant.
scientific article; zbMATH DE number 4046324
scientific article