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Some sufficient conditions for the existence of hyperinvariant subspaces for operators intertwined with unitaries

Volume 246 / 2019

Maria F. Gamal’ Studia Mathematica 246 (2019), 133-166 MSC: Primary 47A15; Secondary 47A60, 47A10. DOI: 10.4064/sm170417-15-1 Published online: 7 September 2018

Abstract

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. These hyperinvariant subspaces are the closures of the range of $\varphi (T)$, where $\varphi $ is a singular inner function if $T$ is polynomially bounded, or $\varphi $ is a function analytic in the unit disc with absolutely summable Taylor coefficients and singular inner part if $T$ is supposed to be only power bounded. Also, an example of a quasianalytic contraction $T$ is given such that the quasianalytic spectral set of $T$ is not the whole unit circle $\mathbb T$, while $\sigma (T)=\mathbb T$. The proofs are based on results by Esterle, Kellay, Borichev and Volberg.

Authors

  • Maria F. Gamal’St. Petersburg Branch
    V. A. Steklov Institute of Mathematics
    Russian Academy of Sciences
    Fontanka 27
    St. Petersburg, 191023, Russia
    e-mail

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