# Tagged Questions

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### Rotating the spectrum of a bounded operator

If $T$ is a bounded operator on a Banach space $X$, and $\sigma(T)$ is its spectrum, what would be an operator whose spectrum is $\sigma(T)$ rotated by $\theta$? For example, $-T$ has as spectrum ...
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### Rudin's proof of invariant subspace existence

I have questions about Rudin's proof of invariant subspace existence. On page 327, point 12.27, How does he get that $Tx=TE(\omega)x$, and How does he know $E(\omega)$ is not the zero map? ...
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### Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
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### renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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### A couple of proofs on a spectrum

Let $T$ be a normal bounded operator. Let ${\lambda}$ be in $({\sigma}(T))$. Without invoking general algebra theories, show that: a) $p({\lambda},{\lambda}^*)$ is in $({\sigma}(T))$ for all ...
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### operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
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### Spectral decomposition

For a compact normal operator, the space can be written as the sum of generalized eigenspaces. So every element can be written as a linear combination of the eigenvectors, one from each eigenspace. ...
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### Spectral decomposition for normal compact operator

My book says $Tx=(\alpha x_{\alpha})$ where the $\alpha$ are eigenvalues the of T. But the image of an operator is not in general a sequence. Do they mean these are the scalars in the linear ...
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### Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...