Let $\{ \lambda_n\}_n \in \ell^\infty(\mathbb{ N})$. 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 $\sum_{ n=1}^{ \infty}\lambda_n (A^*_nx_n\otimes \bar{ B^*_ny_n})$, 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 $\{ \|x_n\|\|y_n\|\}_n \in \ell^\infty(\mathbb{ N})$. We next generalize the classes of Hilbert-Schmidt and trace class operators.
Categories: math.FA