Bounded operators on Hilbert spaces

[Office_Shredder]

There's nothing special about infinite dimensions here. I suggest trying the following: let H and K be one dimensional spaces, pick T and S to be whatever bounded linear functions you want on them. Then write down what $H\oplus K$ and $T \oplus S$ are.

 

[HeinzBor]

$$||T \oplus S (h,k)||^{2} = \sqrt( \langle ((T(h), S(k))) | (T(h), S(k)) \rangle)^{2}$$

 

[HeinzBor]

There's nothing special about infinite dimensions here. I suggest trying the following: let H and K be one dimensional spaces, pick T and S to be whatever bounded linear functions you want on them. Then write down what $H\oplus K$ and $T \oplus S$ are.

Do you mind showing me, I am a bit stuck right now...

 

[PeroK]

Do you mind showing me, I am a bit stuck right now...

The linear functions on $\mathbb R$ for example are of the form $T(x) = ax$, for some constant $a$.

 

[HeinzBor]

The linear functions on $\mathbb R$ for example are of the form $T(x) = ax$, for some constant $a$.

$$|| T \oplus S || = || T, S||$$. $$||T,S||^{2} = ||T||^{2} + ||S||^{2} = ||T \oplus S||^{2}$$And by boundedness of T,S we get

$$||Th, Sk||^{2} = ||Th||^{2} + || Sk ||^{2} \leq C ||h||^{2} + W ||k||^{2}$$... and then I am stuck again.

 

[fresh_42]

Well I am currently in grad school in pure maths coming back from 1 year break from mathematics. I just finished a course in functional analysis and I am now taking a course on operator algebras.

I think you should recap all exercises you got in functional analysis. Your question is rather basic, which in my opinion shows a lack of practice in solving problems.

 

[fresh_42]

... and then I am stuck again.

Maybe you should think from the end. What does it mean that $T\oplus S$ is bounded? Which inequality do you have to end up with?

 

[HeinzBor]

Maybe you should think from the end. What does it mean that $T\oplus S$ is bounded? Which inequality do you have to end up with?

It was mentioned earlier in this thread. $$||T \oplus S (h \oplus k)|| \leq C ||h \oplus k||$$ for $$h \oplus k \in H \oplus K$$

 

[fresh_42]

It was mentioned earlier in this thread. $$||T \oplus S (h \oplus k)|| \leq C ||h \oplus k||$$ for $$h \oplus k \in H \oplus K$$

Yes. But what does this mean?

Let us see what you already have. E.g. we actually have
$$0\leq ||T \oplus S (h \oplus k)|| \leq C ||h \oplus k||$$
This is equivalent to $$0\leq ||T \oplus S (h \oplus k)||^2 \leq C^2 ||h \oplus k||^2.$$
You used equality earlier, which is wrong, but the inequality holds because the root function is strictly increasing. With the hint you got in post #8 we have
$$0\leq ||T \oplus S (h \oplus k)||^2=\|T(h)\|^2+\|S(k)\|^2 \leq C^2 ||h \oplus k||^2.$$
This means we have to show:
$$0\leq ||T \oplus S (h \oplus k)||^2=\|T(h)\|^2+\|S(k)\|^2 \leq \ldots\leq \ldots\leq \ldots \leq C^2 ||h \oplus k||^2.$$
In other words: You have to fill in the gaps. And what is $\|h\oplus k\|^2$?

Note that $C$ is used as a variable, which was your decision in post #38. So whatever you do, I don't want to see something like $\|T(h)\|\leq C\|h\|$.

 

[HeinzBor]

Yes. But what does this mean?

Let us see what you already have. E.g. we actually have
$$0\leq ||T \oplus S (h \oplus k)|| \leq C ||h \oplus k||$$
This is equivalent to $$0\leq ||T \oplus S (h \oplus k)||^2 \leq C^2 ||h \oplus k||^2.$$
You used equality earlier, which is wrong, but the inequality holds because the root function is strictly increasing. With the hint you got in post #8 we have
$$0\leq ||T \oplus S (h \oplus k)||^2=\|T(h)\|^2+\|S(k)\|^2 \leq C^2 ||h \oplus k||^2.$$
This means we have to show:
$$0\leq ||T \oplus S (h \oplus k)||^2=\|T(h)\|^2+\|S(k)\|^2 \leq \ldots\leq \ldots\leq \ldots \leq C^2 ||h \oplus k||^2.$$
In other words: You have to fill in the gaps. And what is $\|h\oplus k\|^2$?

Note that $C$ is used as a variable, which was your decision in post #38. So whatever you do, I don't want to see something like $\|T(h)\|\leq C\|h\|$.

Thanks a lot for your detailed explanation. Okay so what I thought is the following, following your hint:

$$0 \leq ||T \oplus S (h \oplus k)||^{2} = ||T(h)||^{2} + ||S(k)||^{2}\\
\leq || T ||^{2} || h ||^{2} + || S ||^{2} || k ||^{2} \ (using \ boundedness \ of \ T,S)$$
$$\leq max \{ ||T||^{2}, ||S||^{2} \} (||h||^{2}_{H} + ||k||_{K}^{2}) $$
$$= max \{ ||T||^{2}, ||S||^{2} \} ||h \oplus k||^{2}_{H \oplus K}$$
Put $$C:= max \{ ||T||^{2}, ||S||^{2} \} $$, then

$$0 \leq ||T \oplus S (h \oplus k)||^{2} \leq C ||h \oplus k||^{2}_{H \oplus K}$$

 

[fresh_42]

Looks good. Except for a missing +.

 

[HeinzBor]

Looks good. Except for a missing +.

I think it should be fixed now

 

[fresh_42]

I think it should be fixed now

The plus sign is still missing (1st line).

 

[HeinzBor]

The plus sign is still missing (1st line).

ahh yes of course I see it now thanks a lot!