ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 2 (2003)
p. 191 – 196

On the Range and the Kernel of the Elementary operators å AiXBiX
S. Mecheri

Abstract.  Let $B(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. For $% A=(A_{1},A_{2}...A_{n})$ \,and \,$B=(B_{1},B_{2}...B_{n})$ \ $n$-tuples in \,$B(H)$, \,we define the elementary operator $\Delta _{A,B}X:B(H)\mapsto B(H)$ by $% \Delta _{A,B}=\sum A_{i}XB_{i}-X.$ In this paper we show that if $\Delta _{A,B}=0=\Delta _{A,B}^{*},$ then $$\left\| T+\Delta _{A,B}(X)\right\| _{{\mathcal I}}\geq \left\| T\right\| _{{\mathcal I}}$$ for all $X\in {\mathcal I}$ (proper bilateral ideal) and for all $T\in \ker (\Delta _{A,B}\mid {\mathcal I})$.

AMS subject classification:  47B47, 47A30, 47B20; Secondary: 47B10
Keywords:  Elementary operators, Schatten $p$-classes, unitairy invariant norm, orthogonality