Frames and operators in Schatten classes

Abstract

Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\le p<\infty$, $T$ belongs to the Schatten class $S_p$ if and only if $\{\Vert T{f}_n\Vert \}\in {\ell }^p$ for \emph{every} frame $\{f_n\}$ in $H$; and for $0<p\le 2$, $T$ belongs to $S_p$ if and only if $\{\Vert T{f}_n\Vert \}\in {\ell }^p$ for \emph{some} frame $\{f_n\}$ in $H$. Similar conditions are also obtained in terms of the sequence $\{<T{f}_n,f_n>\}$ and the double-indexed sequence $\{<T{f}_n,f_m>\}$.

27 pages