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Martin Sleziak

10:26

in Discussion between levap and Student, 7 secs ago, by Martin Sleziak

@Student I was thinking about adding a bounty to your question By using the spectral radius formula, how to prove that $r(T)\leq \omega(T)$?.

in Discussion between levap and Student, 10 secs ago, by Martin Sleziak

Is that question still unresolved? Feel free to drop me a line in my chat room.

BTW wouldn't something like that be enough to get answer you're satisfied with? Proof of Gelfand formula for spectral radius

I am referring mainly to this comment:

In theorem 1.1 the author used the definition of r(T) with the spetrum but I want to use the above formula..and thank you — Student Aug 11 at 8:28

Perhaps you can find some other posts about the same formula among search using Approach0.

BTW bounty was suggested also by the answerer:

Yes, maybe a bounty will buy someone's time to sort out your reading problems. — uniquesolution yesterday

Student

10:51

You want that I start a bounty?

Martin Sleziak

Oh, you can do so, too.

I thought that you are hesitant to do so, so I offered that I can start a bounty - assuming the question is not already solved and that it is improved a bit.

Student

You have an answer? Thank you

Martin Sleziak

@Student No I do not.

From what you wrote in your comments it seemed that you're close to an aswer.

You wrote:

> In theorem 1.1 the author used the definition of r(T) with the spetrum but I want to use the above formula..and thank you.

Student

No, I don't find an answer

I hope that I get an answer

Martin Sleziak

So it seems that the only thing missing is comparison of two definitions of spectral radius.

Student

10:54

yes

Martin Sleziak

That's why I pointed out this question: Proof of Gelfand formula for spectral radius

Student

I want to use the spectral radius formula

Martin Sleziak

BTW where does the problem in your question come from?

I'll have to leave now, sorry.

Student

Thank you if you can help me to solve the problem.

Martin Sleziak

11:44

@Student And why is the solution using spectrum together with the proof that $r(T)=\sup\{|\lambda|; \lambda \in \sigma(T)\} = \lim\limits_{n\to\infty} \|T^n\|^{1/n}$ not satisfactory?

It seems that it is possible to find this proof in various books. For example, Theorem 2.10 in Kubrusly C.S. Spectral theory of operators on Hilbert spaces.

Student

12:10

Thank you for your answer. If I can prove this result in Hilbert spaces, I can generalized it in semi hilbert space

and in semi Hilbert space, the A spectral radius is given only by the formula in the picture

it is not defined using spectrum as in Hilbert spaces

I hope that you understand me

Martin Sleziak

12:45

@Student So you should probably include in your post that this is the motivation why you're trying to do this using this formula (and not eigenvalues).

Student

ok thank you

Martin Sleziak

If you edit your question in a way which explains that this is the motivation, maybe I'll offer the bounty.

Student

You can edit my question. Perhaps you can explain better than me and thank you

Martin Sleziak

@Student No, I can't explain it better than you. After all, it is your question and you know better what you are actually trying to achive.

Student

ok thank you .

3 hours later…

Student