Re: Spectral Radius question


Tom Smith wrote:
David C. Ullrich wrote:
On Fri, 19 Jan 2007 18:12:01 +0000, Tom Smith <ts336@xxxxxxxxx> wrote:

I need a hint for the following question (Bollobas, Linear Analysis, 12.8):

"Let S, T be bounded linear operators on X. Show that if S and T commute then

r(S + T) <= r(S) + r(T)
r(ST) <= r(S) + r(T)."

(here r(S) = sup { l : (T-lI) is not invertible } is the spectral radius).

Using both "l" and "I" is a very bad idea - they look more or less
identical in many fonts!


Noted: thanks.

I can see that the second inequality follows immediately from the Gelfand
formula r(T) = lim ||T^n||^(1/n).

Or rather it would follow if you fixed the typo.


I honestly can't see a typo here - what is it? This is copied direct from the
textbook.

I believe the typo is connected, not with the expression of spectral
radius by the power formula, but with the inequality you gave on
r(ST). Consider S = T = 3I. What would your inequality claim?

I assume the first one is done the same way
but can't see how. Can anyone give me a (gentle) prod in the right direction?

Do you know anything about commuttative Banach algebras? (Complex
homomorphisms/maximal ideals...)


Only the definitions: nothing else is covered in the course I'm taking (though
we rather suspect the lecturer didn't get through all the material he wanted).
Can it not be done direct from the definition or the formula?


Tom

.

Relevant Pages

  • Re: Spectral Radius question
    ... David C. Ullrich wrote: ... T be bounded linear operators on X. Show that if S and T commute then ... identical in many fonts! ...
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  • Re: Spectral Radius question
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  • Re: Spectral Radius question
    ... T be bounded linear operators on X. Show that if S and T commute then ... identical in many fonts! ...
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