Size of the peripheral point spectrum under power or resolvent growth conditions

Abstract

We characterize Jamison sequences, that is sequences (n k ) of positive integers with the following property: every bounded linear operator T acting on a separable Banach space with sup k T n k < +∞ has a countable set of peripheral eigenvalues. We also discuss partially power-bounded operators acting on Banach or Hilbert spaces having peripheral point spectra with large Hausdorff dimension. For a Lavrentiev domain Ω in the complex plane, we show the uniform minimality of some families of eigenvectors associated with peripheral eigenvalues of operators satisfying the Kreiss resolvent condition with respect to Ω. We introduce and study the notion of Ω-Jamison sequence, which is defined by replacing the partial power-boundedness condition sup k T n k < +∞ by sup k F Ω n k (T ) < +∞, where F Ω n is the nth Faber polynomial of Ω. A characterization of Ω-Jamison sequences is obtained for domains with sufficiently smooth boundary.

Hilbertian Jamison sequences and rigid dynamical systems

“…A complete characterization of Jamison sequences was obtained in [2]. It is formulated as follows: It is well known that ϕ is weakly mixing if and only if ϕ × ϕ is an ergodic transformation of X × X endowed with the product measure μ × μ.…”

“…There exists a bounded linear operator T on the Hilbert space 2 (N) such that T has perfectly spanning unimodular eigenvectors and sup k 0…”

“…This can be rewritten using the distance on T defined by d (n p ) (λ, μ) = sup p 0 λ n p − μ n p as: for any ε > 0 there exists a λ ∈ T \ {1} such that d (n p ) (λ, 1) ε. This means (see [2]) that there exists an uncountable subset K of T such that (K, d (n p ) ) is a separable metric space. Thus K contains a subset K which is uncountable and perfect for the distance d (n p ) .…”

“…Then we take λ 2 in K such that d (n p ) (λ 2 , λ j (2) ) is extremely small, which is possible since λ j (2) = λ 1 is not isolated in K .…”

Faber-hypercyclic operators

“…An interesting feature in this study is the influence of the geometry of the domain Ω and of the smoothness of its boundary on Ω-hypercyclicity. For another instance of such a relationship between the geometry of the domain and the behaviour of the Faber polynomials of an operator (in relation to the boundary point spectrum σ p (T ) ∩ ∂Ω), see [2]. If H + (T ) and H − (T ) are dense in X, then T is hypercyclic.…”