norm of a matrix that its entries are operators in B(H).

Question

Let S is a subset of B(H). Define $M_2(S)=\{T= \left( \begin{array}{ccc} A & B \\ C & D \\ \end{array} \right) : A,B,C,D \in S\}$.

what is the relationship between $||T||$ and ||A||,||B||,||C|| and ||D||?

thanks for your help.

Answer 1

I am assuming that you are working with the operator norm on $B(H)$ and with the unique $C^*$-norm on $M_2(B(H))$ given by identifying this space with $B(H \oplus H)$.

The short answer is that in general there is no equation in the norms of $A$, $B$, $C$, and $D$ that results in $\Vert T \Vert$.

To see this, we can consider the case where $H= \mathbb C$ and so we are working with $M_2(\mathbb C)$.

The matrix $\left(\begin{array}{cc}1 & -1 \\1 & 1\end{array}\right)$ has norm $\sqrt 2$ while the matrix $\left(\begin{array}{cc}1 & 1 \\1 & 1\end{array}\right)$ has norm $2$. So the norm of $T$ does not only depend on the norms of $A, B, C$ and $D$.

One can, of course, easily come up with some inequalities that bound $\Vert T \Vert$.

If $B=C=0$, then $\Vert T \Vert = \max \{ \Vert A \Vert, \Vert D \Vert \}$.

If $A=D=0$, then $\Vert T \Vert = \max \{ \Vert B \Vert, \Vert C \Vert \}$.

As a result, $\Vert T \Vert \leq \max \{ \Vert A \Vert, \Vert D \Vert \} + \max \{ \Vert B \Vert, \Vert C \Vert \}$.

In response to comment:

The norm of $T=\left(\begin{array}{cc}0 & A \\B & 0\end{array}\right)$ is the square root of the norm of $$ \left(\begin{array}{cc}0 & A \\B & 0\end{array}\right)\left(\begin{array}{cc}0 & B^* \\A^* & 0\end{array}\right) = \left(\begin{array}{cc}AA^* & 0 \\0 & BB^*\end{array}\right).$$

So $\Vert T \Vert^2 = \max \{ \Vert A \Vert ^2, \Vert B \Vert^2 \}$.