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Spectral mapping theorem is as follows:
https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf

Is Spectral mapping theorem true for point spectrum ?

user107723
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1 Answers1

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Your reference is talking about matrices. Which spectral mapping theorem are you interested in? For example, here's one version (see e.g. Rudin, "Functional Analysis", theorems 10.28 and 10.33):

Suppose $T$ is a bounded linear operator on the complex Banach space $X$, $\Omega \subseteq \mathbb C$ open with $\sigma(T) \subset \Omega$, and $f$ analytic in $\Omega$. Then the holomorphic functional calculus defines $f(T)$ such that, among other things (where $\sigma_p$ denotes point spectrum):

  1. $f(\sigma(T)) = \sigma(f(T))$
  2. $f(\sigma_p(T)) \subseteq \sigma_p(f(T))$
  3. If there is no connected component of $\Omega$ on which $f$ is constant, then $f(\sigma_p(T)) = \sigma_p(f(T))$.

However, without that restriction the statement would be false. Consider e.g. an operator $T$ which has no point spectrum, and let $f$ be identically $0$. Then $\sigma_p(f(T)) = \sigma_p(0) = \{0\}$, but $f(\sigma_p(T)) = \emptyset$.

Robert Israel
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    If $f$ is a polynomial of degree $n$ of complex coefficient, then does $(3)$ hold ? – user107723 Mar 11 '14 at 05:19
  • Sorry I meant to say, If $f$ is a polynomial of degree $n$ of complex coefficient, then does $(3)$ hold without the "connected" condition? – user107723 Mar 11 '14 at 05:32
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    If $n \ge 1$, $f$ is not constant on any open set, so yes. A polynomial of degree $0$ is constant. – Robert Israel Mar 11 '14 at 06:34
  • It is true that $f$ is not constant in an open set if $f$ is a polynomial. But how can we be sure that $f$ is not constant in a connected set ? Because every connected set is not open. – user107723 Mar 11 '14 at 06:56
  • $\Omega$ is open, so its connected components are open. A polynomial of degree $n \ge 1$ takes any given value in at most $n$ different points. – Robert Israel Mar 11 '14 at 14:59
  • If $A$ and $B$ are two bounded linear operator on a Banach space, whose spectrum and point spectrum are known, then what can we say about the spectrum and point spectrum of $A+B$ and $AB$ – user107723 Mar 31 '14 at 05:58
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    This should be a new question, not a comment – Robert Israel Mar 31 '14 at 14:58