  This paper is a continuation of a recent work on a new norm, christened the $
(\alpha, \beta)$-norm, on the space of bounded linear operators on a Hilbert
space. We obtain some upper bounds for the said norm of $n\times n$ operator
matrices. As an application of the present study, we estimate bounds for the
numerical radius and the usual operator norm of $n\times n$ operator matrices,
which generalize the existing ones.
