Consider the block-operator matrix $$ D = \begin{bmatrix} D^{1,1} & D^{1,2} & \dots & D^{1,M} \\ D^{2,1} & D^{2,2} & \dots & D^{2,M} \\ \vdots & \vdots & \ddots & \vdots \\ D^{M,1} & D^{M,2} & \dots & D^{M,M} \end{bmatrix}, $$ where each block $D^{m,n}$ is an operator $D^{m,n}:L^2(\mathbb{R}) \to L^2(\mathbb{R})$.
What norm can we use for $D$? For example, can we define a norm here by simply taking the 'largest' block, i.e., $$ ||D|| = \max_{p,q \in M} ||D^{p,q}||_{L^2(\mathbb{R})\to L^2(\mathbb{R})}. $$
Is this a valid norm?