We study hypercyclicity of the Toeplitz operators in the Hardy space H 2 (D) with symbols of the form p(z)+ ϕ(z), where p is a polynomial and ϕ ∈ H ∞ (D). We find both necessary and sufficient conditions for hypercyclicity which almost coincide in the case when deg p = 1.1991 Mathematics Subject Classification. 47A16, 47B35, 30H10.
Recently, Sophie Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. We give a similar construction using a functional model for rank one perturbations of singular unitary operators.Mathematics Subject Classification. 47A16, 30A76, 30H10.
In this note we discuss an open problem whether a truncated Toeplitz operator on a model space can be hypercyclic. We compute point spectrum and eigenfunctions for a class of truncated Toeplitz operators with polynomial analytic and antianalytic parts. We show that, for a class of model spaces, truncated Toeplitz operators with symbols of the form Φ(z) = az +b+cz, |a| = |c|, are not hypercyclic.
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