Yesterday I posted this one regarding direct sum on Hilbert spaces $H_1$ and $H_2$ - have a look!
Direct sum of two Hilbert spaces is a inner product.
I am studying bounded operators and I just want to know if we can say something about the norm of my direct sum. I.e. let $T_1\in B(H_1)$ and $T_2\in B(H_2)$ then $T_1\oplus T_2$ is bounded by using the definition of bounded operator but how about its norm? I cannot se how this should be possible. Any suggestion? I belive we should use operator norm but I still cannot see how that would work. Thanks in advance.
For any $(x,y)\in H_1\oplus H_2$, by definition, $T_1\oplus T_2 (x,y)=(T_1 x,T_2 y$). Thus \begin{align*}\|T_1\oplus T_2(x,y)\|=\sqrt{\|T_1x\|_1^2+\|T_2y\|_2^2}&\leq \sqrt{\|T_1\|_1^2\|x\|_1^2+\|T_2\|_2^2\|y\|_2^2}\\&\leq\max\{\|T_1\|_1,\|T_2\|_2\}\|(x,y)\|.\end{align*} This gives an upper bound for what the norm of $T_1\oplus T_2$ is (and shows that it is definitely bounded at least). To show that this bound is the norm, I leave you the fun of choosing an appropriate $(x,y)$.