  We address the problems of computing operator norms of matrices induced by
given norms on the argument and the image space. It is known that aside of a
fistful of "solvable cases," most notably, the case when both given norms are
Euclidean, computing operator norm of a matrix is NP-hard. We specify rather
general families of norms on the argument and the images space ("ellitopic" and
"co-ellitopic," respectively) allowing for reasonably tight computationally
efficient upper-bounding of the associated operator norms. We extend these
results to bounding "robust operator norm of uncertain matrix with box
uncertainty," that is, the maximum of operator norms of matrices representable
as a linear combination, with coefficients of magnitude $\leq1$, of a
collection of given matrices. Finally, we consider some applications of norm
bounding, in particular, (1) computationally efficient synthesis of affine
non-anticipative finite-horizon control of discrete time linear dynamical
systems under bounds on the peak-to-peak gains, (2) signal recovery with
uncertainties in sensing matrix, and (3) identification of parameters of time
invariant discrete time linear dynamical systems via noisy observations of
states and inputs on a given time horizon, in the case of
"uncertain-but-bounded" noise varying in a box.
