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 $\Delta {A,B}(X)=\sum{i=1}^{n}A{i}XB\_{i}-X.$ In this paper we minimize the map $F\_{p}(X)= \left| T -\Delta \_{A,B}(X) \right| \_{p}^{p}$, where $T\in \ker\Delta {A,B}\cap C{p}$, and we classify its critical points.