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

Abstract: 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.