Frames and operators in Schatten classes (Q2786838)

The paper is devoted to study in a Hilbert space \(H\) the characterization of operators \(T\) from the Schatten class \(S_{p}\) for \(0<p<\infty\) by the sequence \(\left\{ \left\| Tf_{n}\right\| \right\}\) for any frame \(\left\{f_{n}\right\}\) in \(H\). It is proved that, for \(2 \leq p< \infty\), \(T \in S_{p}\) is equivalent to \(\left\{ \left\| Te_{n}\right\| \right\} \in l_{p}\) for any orthonormal basis \(\left\{e_{n}\right\}\) or, \(\left\{ \left\| Tf_{n}\right\| \right\} \in l_{p}\) is true for any frame. However, this does not hold if \(\left\{ \left\| Tf_{n}\right\| \right\} \in l_{p}\) for some frame \(\left\{f_{n}\right\}\) in \(H\), namely, it is shown that when \(2<p<\infty\) and \(\varepsilon>0\), for any \(T \in S_{p+\varepsilon}\setminus S_{p}\) there exists a frame \(\left\{f_{n}\right\}\) in \(H\) such that \(\left\{ \left\| Tf_{n}\right\| \right\} \in l_{p}\) and for any orthonormal basis \(\left\{e_{n}\right\}\) in \(H\) there exists \(S \in S_{p+\varepsilon}\setminus S_{p}\) such that \(\left\{ \left\| Sf_{n}\right\| \right\} \in l_{p}\). It is established that these results do not carry over the case of \(0<p<2\). It is proved in this case that the condition \(T \in S_{p}\) is equivalent to fulfilling \(\left\{ \left\| Tf_{n}\right\| \right\} \in l_{p}\) for some orthonormal basis \(\left\{e_{n}\right\}\) and some frame \(\left\{f_{n}\right\}\) in \(H\). When \(0<p<1\), for any operator \(T\) in \(H\) there exists a frame \(\left\{f_{n}\right\}\) such that \(\left\{ \left\| Tf_{n}\right\| \right\} \notin l_{p}\) , and also for any orthonormal basis \(\left\{e_{n}\right\}\) in \(H\) there exists \(S \in S_{p}\) such that \(\left\{ \left\| Sf_{n}\right\| \right\} \notin l_{p}\). The obtained results are applied to some operator in the space \(A^{2}\) of analytic functions on the unit disc \(D\) .