Some Norm Inequalities for Operators

Kittaneh, Fuad

Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 87-96

Résumé

Let ${{A}_{i}},\,{{B}_{i}}$ and ${{X}_{i}}\,(i\,=\,1,\,2,\ldots ,\,n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty\right)$ which satisfy the relation $f\,(t)g(t)\,=\,t$ for all $t$ in $\left[ 0,\infty\right)$ , then $${{\left\| \left| \,{{\left| \sum\limits_{i=1}^{n}{A_{i}^{*}{{X}_{i}}{{B}_{i}}} \right|}^{r}} \right| \right\|}^{2}}\,\le \,\left\| \left| {{\left( \sum\limits_{i=1}^{n}{A_{i}^{*}f{{(\left| X_{i}^{*} \right|)}^{2}}{{A}_{i}}} \right)}^{r}} \right| \right\|\,\,\left\| \left| {{\left( \sum\limits_{i=1}^{n}{B_{i}^{*}g{{(\left| {{X}_{i}} \right|)}^{2}}\,{{B}_{i}}} \right)}^{r}} \right| \right\|$$ for every $r\,>\,0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Détail

DOI : 10.4153/CMB-1999-010-6

Mots-clés : 47A30, 47B10, 47B15, 47B20, unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequality