Similarity to a contraction, for power-bounded operators with finite peripheral spectrum

deLaubenfels, Ralph

doi:10.1090/S0002-9947-98-02303-4

Similarity to a contraction, for power-bounded operators with finite peripheral spectrum
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Trans. Amer. Math. Soc. 350 (1998), 3169-3191 Request permission

Abstract:

Suppose $T$ is a power-bounded linear opertor on a Hilbert space with finite peripheral spectrum (spectrum on the unit circle). Several sufficient conditions are given for $T$ to be similar to a contraction. A natural growth condition on the resolvent in half-planes tangent to the unit circle at the peripheral spectrum is shown to be equivalent to $T$ having an $H^\infty (\mathcal P)\cap C(\overline {\mathcal P})$ functional calculus, for some open polygon $\mathcal P$ contained in the unit disc, which, in turn, is equivalent to $T$ being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of $T$ be square summable also implies that $T$ is similar to a contraction.

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