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0answers
39 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...
1
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0answers
46 views

What is a good book that focuses on the applications of complex analysis and spectral theory?

My research involves a great deal of complex analysis and spectral theory, and I always feel a bit flustered when non mathematicians ask me what I study. It's hard to explain the math in layman's ...
1
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1answer
19 views

regarding a proof of $\|\theta(e^{i\lambda})\|^2$

When studying the spectral representation of time series, I read the following formula, I am not clear how to prove the second equation. I expand the left side of the second equation with the ...
1
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1answer
56 views

Showing a bound on a contour integral

I'm working through M. Schechter's 'Principles of Functional Analysis' and I'm working through a proof on page 136 that shows that the spectral radius $r_{\sigma} (T) $ of a bounded linear operator ...
1
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1answer
65 views

Proof that $A^n = \frac{1}{2\pi i} \oint _C z^n \left(z - A \right)^{-1} dz$

I'm working my way through Martin Schechter's 'Principles of Functional Analysis' (2nd ed.) and am trying to understand his proof of the following theorem, given on page 136: "Let $T:X\to X$ be any ...
0
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0answers
74 views

Spectral theory - a simple application

I'm following Martin Schechter's 'Principles of Functional Analysis' (Second Edition, 2002) and am interested in the spectral theory chapter (chapter six). In particular, I wish to make us of ...
2
votes
1answer
237 views

Simple isolated eigenvalue and pole of the resolvent

Let $T$ be bounded linear operator on some complex Banach space, and $\lambda$ an eigenvalue of $T$ which is isolated in its spectrum, and such that $\bigcup_{n\ge 1} N((T- \lambda I)^n)$ is ...
4
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0answers
143 views

When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...