JournalszaaVol. 23 , No. 2DOI 10.4171/zaa/1200

Another Version of Maher's Inequality

  • Salah Mecheri

    King Saud University, Riyadh, Saudi Arabia
Another Version of Maher's Inequality cover

Abstract

Let HH be a separable infinite dimensional complex Hilbert space, and let L(H)L(H) denote the algebra of bounded linear operators on HH into itself. Let A=(A1,A2...,An)A=(A_{1},A_{2}...,A_{n}), B=(B1,B2...,Bn)B =(B_{1},B_{2}...,B_{n}) be n-tuples of operators in L(H)L(H). We define the elementary operator ΔA,B:L(H)L(H)\Delta _{A,B}: L(H) \mapsto L(H) by ΔA,B(X)=i=1nAiXBiX.\Delta _{A,B}(X)=\sum_{i=1}^{n}A_{i}XB_{i}-X. In this paper we minimize the map Fp(X)=TΔA,B(X)ppF_{p}(X)= \left\| T -\Delta _{A,B}(X) \right\| _{p}^{p}, where TkerΔA,BCpT\in \ker\Delta _{A,B}\cap C_{p}, and we classify its critical points.