Question: Consider the Hilbert space $ H=\ell^2(\mathbb{Z})$. Let ${\rm L}(H)$ be the set of all linear operators on $H$ onto itself. Find a norm $N$ and a domain $DN\subset {\rm L}(H)$ for $N$ satisfying simultaneously the following three conditions: (i) $DN$ is convex; (ii) $DN \ni I$ (identity operator) and (iii) $N$ is strictly convex. That is, $N( (1 - t)x + ty ) < (1 - t) N(x) + tN(y)$ for any $t\in (0,1)$ and $x,y\in DN$ with $x\neq y$.

In affirmative case, what is the largest $DN$ we can take?