  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.
