Quasianalytic polynomially bounded operators

Abstract

Quasianalytic contractions form the crucial class in the quest for proper invariant and hyperinvariant subspaces for asymptotically non-vanishing Hilbert space contractions. The property of quasianalycity relies on the concepts of unitary asymptote and $H^{ı}$-functional calculus. These objects can be naturally defined in the setting of polynomially bounded operators too, which makes possible to extend the study of quasianalycity from contractions to this larger class. Carrying out this program we pose also several interesting questions.

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• On polynomially bounded operators with shift-type invariant subspaces
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• On similarity to contractions of class $C_{\cdot 0}$ with finite defects
• Example of quasianalytic contraction whose spectrum is a proper subarc of the unit circle