Another Version of Maher's Inequality

Abstract

Let $H$ be a separable infinite dimensional complex Hilbert space, and let $L(H)$ denote the algebra of bounded linear operators on $H$ into itself. Let $A=(A_1,A_2...,A_n)$, $B=(B_1,B_2...,B_n)$ be n-tuples of operators in $L(H)$. We define the elementary operator ${\Delta }_{A,B}:L(H)\mapsto L(H)$ by ${\Delta }_{A,B}(X)={\sum }_{i=1}^n{A}_{i}X{B}_i-X.$ In this paper we minimize the map $F_p(X)={\Vert T-{\Delta }_{A,B}(X)\Vert }_p^p$, where $T\in \ker {\Delta }_{A,B}\cap C_p$, and we classify its critical points.