which norms can be realized as operator norms?

Question

Assume $(V,∥∥_V),(W,∥∥_W)$ are both finite dimensional normed spaces. We have the induced operator norm on ${\rm Hom}(V,W)$.

It turns out that the operator norm is induced by an inner product iff both $V,W$ are inner product spaces and at least one of $V,W$ has dimension 1. (This is proved here).

Are there any other obstructions for a norm on ${\rm Hom}(V,W)$ to be realized as an operator norm for some suitable norms on $V,W$?

(The main interest is in the case where $\dim V>1,\dim W>1$. Take a norm on the Hom space which does not come from an inner product. Is it realizable?)

Answer 1

Operator norms are the same thing as injective tensor norms or, equivalently, smallest dual cross norms on $\operatorname{Hom}(V, W) \simeq V^\ast \otimes W$. This means that the dual norm $\Vert \cdot \Vert^\ast$ on $V \otimes W^\ast$ has to satisfy the following:

$$\Vert x \Vert^\ast = \inf_{x = \sum_i y_i \otimes z_i} \sum_i \Vert y_i \Vert \Vert z_i \Vert$$

where the infimum is over all possible representations of a tensor as a combination of rank-ones. This follows immediately from the definition of the operator norm, i.e. that it's determined by testing against rank-ones.

In particular, the unit ball of $\Vert \cdot \Vert^\ast$ is the convex hull of its rank-one vectors. Thus, for example, it cannot be strictly convex (unless everything is of rank one, which means that $\min(\dim V, \dim W) = 1$). This generalizes the Euclidean case.

Answer 2

Here is a non trivial constraint : ${\rm Hom}(V,W)$ contains (is spanned by) rank one morphisms $$v\mapsto\ell(v)w,\qquad\ell\in V',w\in W.$$ If a given norm over ${\rm Hom}(V,W)$ is induced, then $$\|w\otimes\ell\|=\|w\|_W\|\ell\|_*.$$ This yields the necessary condition $$\|w_1\otimes\ell_1\|\cdot\|w_2\otimes\ell_2\|=\|w_1\otimes\ell_2\|\cdot\|w_2\otimes\ell_1\|,\qquad\forall w_1,w_2\in W,\ell_1,\ell_2\in V'.$$ This raises the question whether this is the only restriction.