- #1

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- Homework Statement:
- Bounded operators on hilbert spaces direct sum.

- Relevant Equations:
- operators

I have to show that for two bounded operators on Hilbert spaces ##H,K##, i.e. ##T \in B(H)## and ##S \in B(K)## that the formula ##(T \bigoplus S) (\alpha, \gamma) = (T \alpha, S \gamma)##, defined by the linear map ##T \bigoplus S: H \bigoplus K \rightarrow H \bigoplus K ## is bounded. After that I must compute the norm of ##T \bigoplus S## in terms of the norms of ##T## and ##S##.

Furthermore, it is given that the direct sum ##H \bigoplus K## is the vector space ##H \times K## with the standard coordinate vector space structure. with the inner product given by

$$\langle (x,y) | (x',y') \rangle := \langle x | x' \rangle_{H} + \langle y | y' \rangle_{K}\, , \,(x,y), (x',y') \in H \times K$$.

However, I am not sure if that is of any use in this problem.

What I have done so far:

We say that an operator ##A: H \rightarrow K## is bounded if there exists ##C > 0## such that ##||Ah|| \leq C ||h|| \forall h \in H##.

So naturally what I must show in the first part is that ##||(T \bigoplus S) \Gamma|| \leq C ||\Gamma|| \ \forall \Gamma \in (T \bigoplus S)##. At this point I am a bit unsure of how this element ##\Gamma## looks like, hence the notation. About the "norm-part" of the question I believe that since both ##T## and ##S## are bounded operators I have to use the fact that

\begin{align*}

\|T\|=\sup \{ \frac{\|Ah\|}{ \|h\|} : h \neq 0 \}.

\end{align*}

The only way of thinking how to solve this problem would be to somehow show that

##\|TS\| \leq |\|T\| \|S\|##, which should be fairly simple with the norm I just mentioned. But to me that does not seem right because in that regard I don't show anything about the direct sum of the two bounded operators. I am really stuck any help is appreciated.

Furthermore, it is given that the direct sum ##H \bigoplus K## is the vector space ##H \times K## with the standard coordinate vector space structure. with the inner product given by

$$\langle (x,y) | (x',y') \rangle := \langle x | x' \rangle_{H} + \langle y | y' \rangle_{K}\, , \,(x,y), (x',y') \in H \times K$$.

However, I am not sure if that is of any use in this problem.

What I have done so far:

We say that an operator ##A: H \rightarrow K## is bounded if there exists ##C > 0## such that ##||Ah|| \leq C ||h|| \forall h \in H##.

So naturally what I must show in the first part is that ##||(T \bigoplus S) \Gamma|| \leq C ||\Gamma|| \ \forall \Gamma \in (T \bigoplus S)##. At this point I am a bit unsure of how this element ##\Gamma## looks like, hence the notation. About the "norm-part" of the question I believe that since both ##T## and ##S## are bounded operators I have to use the fact that

\begin{align*}

\|T\|=\sup \{ \frac{\|Ah\|}{ \|h\|} : h \neq 0 \}.

\end{align*}

The only way of thinking how to solve this problem would be to somehow show that

##\|TS\| \leq |\|T\| \|S\|##, which should be fairly simple with the norm I just mentioned. But to me that does not seem right because in that regard I don't show anything about the direct sum of the two bounded operators. I am really stuck any help is appreciated.

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