Multipliers for operator-valued Bessel sequences, generalized Hilbert-Schmidt and trace classes

Let $\{{\lambda }_n{\}}_n\in {\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}\lambda_n (x_n\otimes \bar{y_n})$,where${x_n}_n$,${y_n}_n$ are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \cite{BALAZS3} generalized this by replacing $\{x_n{\}}_n$ and $\{y_n{\}}_n$ by Bessel sequences. In this paper, we generalize this by studying the operators of the form ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n(A_n^{\ast }{x}_n\otimes \overline{B_n^{\ast }{y}_n})$, where $\{A_n{\}}_n$ and $\{B_n{\}}_n$ are operator-valued Bessel sequences and $\{x_n{\}}_n$, $\{y_n{\}}_n$ are sequences in the Hilbert space such that $\{\mathrm{\parallel }{x}_n\mathrm{\parallel }\mathrm{\parallel }{y}_n\mathrm{\parallel }{\}}_n\in {\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{N})$. We next generalize the classes of Hilbert-Schmidt and trace class operators.