Let $H$ be a Hilbert space and $B(H)$ be the bounded linear operator on $H$, for $T\in B(H)$, if $T=\left(\begin{array}{ccc} 0 & B \\ A & 0 \\ \end{array}\right)$ on $H=M\oplus M^{\bot}$, then how to compute the norm of $T$? Or, is there any relationship between $||T||$ and $||A||, ||B||$?
what about $T=\left(\begin{array}{ccc} A & B \\ C & D \\ \end{array}\right)$?
In the first case, $$ \left\|T\begin{bmatrix}h\\ k\end{bmatrix}\right\|=\left\|\begin{bmatrix}Bk\\ Ah\end{bmatrix}\right\|. $$ As we are free to choose $h$ and $k$, $\|T\|=\max\{\|A\|,\|B\|\}$.
In the general case, I don't think you can say anything.