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?)