Frames and operators in Schatten classes

Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\le p<\mathrm{\infty }$, $T$ belongs to the Schatten class $S_p$ if and only if $\{\mathrm{\parallel }T{f}_n\mathrm{\parallel }\}\in {\mathrm{\ell }}^p$ for \emph{every} frame $\{f_n\}$ in $H$; and for $0<p\le 2$, $T$ belongs to $S_p$ if and only if $\{\mathrm{\parallel }T{f}_n\mathrm{\parallel }\}\in {\mathrm{\ell }}^p$ 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 $\{<T{f}_n,f_m>\}$.