The problem of smallness for operator blocks in \(L_ p\) spaces
Let 1\(\leq p\leq \infty\), e be a measurable subset in [0,1], \(Q_ e\) be the operator of multiplication with the indicator of e. It was known, that \(\inf \{\| Q_ eAQ_ f\|_{L_ p\to L_ p}:\quad me,mf>0\}=0\) for \(L_ 1\) and C and that there exists an operator A in \(L^ 2\) such that \(\| Q_ eAQ_ f\|_{L_ 1\to L_ 2}=1\quad for\quad all\quad e,f\subset [0,1];\quad me,mf>0.\) Theorem 1. Let \(1<p<\infty\). There exists a bounded linear operator A in \(L_ p\), such that \(\| Q_ eAQ_ f\|_{L_ p\to L_ p}\geq 1\quad for\quad all\quad e,f\subset [(0,1];\quad me,mf>0.\) The constructed example of an operator depends on p. The technique of proof is based on earlier results of the firstnamed author about the Haar system.
scientific article; zbMATH DE number 3863909