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

Abstract

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

This is Studia Math., 246(2019), 133-166 enlarged in the following sense: all proofs of lemmas deleted are preserved here

References in corpus (3)

• Quasianalytic polynomially bounded operators
• Invertibility of functions of operators and existence of hyperinvariant subspaces
• Quasianalytic n-tuples of Hilbert space operators

Cited by in corpus (2)

• Invertibility of functions of operators and existence of hyperinvariant subspaces
• Example of quasianalytic contraction whose spectrum is a proper subarc of the unit circle