ACTA issues

On the norm of elementary operators in standard operator algebras

Ameur Seddik

Acta Sci. Math. (Szeged) 70:1-2(2004), 229-236

Abstract. Let ${\cal A}$ be a complex normed algebra. For $A,B\in{\cal A}$, define a basic elementary operator $M_{A,B}\colon{\cal A} \rightarrow{\cal A}$ by $M_{A,B}(X)=AXB$. Given a standard operator algebra ${\cal A}$ acting on a complex normed space and $A,B\in{\cal A}$ we have: (i) The lower estimate $ \|M_{A,B}+M_{B,A} \|\geq2(\sqrt{2}-1) \|A \|\|B \|$ holds. (ii) The lower estimate $ \|M_{A,B}+M_{B,A} \|\geq\|A \|\|B \|$ holds if $$\inf_{\lambda\in C} \|A+\lambda B \|= \|A \|\hbox{ or } \inf_{\lambda\in C} \|B+\lambda A \|= \|B \|.$$ (iii) The equality $ \|M_{A,B}+M_{B,A} \|=2 \|A \|\|B \|$ holds if $$ \|A+\lambda B \|= \|A \|+ \|B \|\hbox{ for some unit scalar }\lambda.$$ These results extend analogous estimates established earlier for standard operator subalgebras of Hilbert space operators.

AMS Subject Classification (1991): 46L35, 47L35, 47B47

Keyword(s): Lower estimate, standard operator algebra, elementary operator

Received August 21, 2003, and in revised form November 11, 2003. (Registered under 5810/2009.)