ON p-CLASS A(s; t) OPERATORS

Abstract

Let $T$ be a bounded linear operator on a complex Hilbert space $H$ and let $T=U|T|$ be the polar decomposition of $T$. Let $s,t>0$ and $0<p\leq 1$.  $T $ is called

 $p$-class $wA(s,t)$ if $(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{tp}{s+t}}\geq |T^{*}|^{2tp}$ and

$(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{sp}{s+t}}\leq |T|^{2sp}$. Also, $T$ is called $p$-class $A(s,t)$ if $(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{tp}{s+t}}\geq |T^{*}|^{2tp}$. We study some properties of $p$-class $wA(s,t)$ operators. Also, we prove tensor product $T\otimes S$ is $p$-class $wA(s,t)$ if and only if $T$ and $S$ are $p$-class $wA(s,t)$.