Let $T$ be a (bounded) self-adjoint operator on a Hilbert space. Is it true that $||T^k|| = ||T||^k$ for all positive integers $k$? It's true for $k=1,2$, and I'm wondering if this could be generalized. I tried this with some examples and it appears to hold at least for these examples, but I can't prove it in general.
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Yes: for any normal operator $\|T\| = \max \{|\lambda|: \; \lambda \in \sigma(T)\}$, where $\sigma(T)$ is the spectrum of $T$, and $\sigma(f(T)) = f(\sigma(T))$ for any continuous function $f$.
Robert Israel
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I haven't seen these claims before (except the latter for polynomial f which should be enough for this case); do you have a proof for them? – Math Maniac Mar 02 '16 at 19:01
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It's all part of the circle of ideas connected to the Spectral Theorem. Almost any good functional analysis text that does the Spectral Theorem for bounded normal operators will have these in some form. I'd look it up in Reed and Simon or Rudin, but I don't have them with me at the moment. – Robert Israel Mar 02 '16 at 20:22