Given some operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \ , \quad \mbox{ for any g} \in V \ . $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?