Generic properties and a criterion of an operator norm

In this paper, we carefully examine the structure of the gradient of an operator norm on a finite-dimensional matrix space. In particular, we derive concise and useful representations for an operator norm and its subgradient, which refine existing results in this area of study. We further use the derived representations to formulate and prove a criterion of an operator norm, the first of its kind, to the best of our knowledge. It essentially states that a matrix norm is an operator norm if and only if the set of its gradients is the set of the outer products of vectors from each pair of the Cartesian product of two vector sets. We also provide several handy tests, based on this criterion, which in certain cases help to determine whether a matrix norm is an operator norm or not. In addition, we generalize our theoretical developments to higher dimensions, i.e. for injective norms on tensor spaces.