[Paper Review] On the intersection of the spectrum of frequently hypercyclic operators with the unit circle

Hans-Peter Beise|arXiv (Cornell University)|Dec 28, 2013

Holomorphic and Operator Theory14 references3 citations

TL;DR

This paper proves that frequently hypercyclic operators on complex Banach spaces cannot have a spectrum contained in the closed unit disc with only finitely many unimodular spectral points under specific linear independence conditions on the spectral arguments. Using tools from complex analysis—particularly Kronecker's theorem and properties of entire functions of exponential type—it establishes that such spectral configurations lead to contradictions with the dynamics of translation operators, extending Shkarin's result on isolated spectral points.

ABSTRACT

We exclude the existence of frequently hypercyclic operators that have a spectrum contained in the closed unit disc and that intersects the unit circle in only finitely many points under certain additional conditions. This extends a result of S. Shkarin, which states that the spectrum of a frequently hypercyclic operator cannot have isolated points.

Motivation & Objective

• To extend Shkarin's result that the spectrum of a frequently hypercyclic operator cannot have isolated points.
• To investigate whether finite sets of unimodular spectral points on the unit circle can coexist with frequent hypercyclicity under additional number-theoretic conditions.
• To establish necessary spectral conditions for frequent hypercyclicity by linking operator dynamics to the zero sets of entire functions of exponential type.
• To resolve a potential contradiction with earlier claims in the literature by showing that certain spectral configurations are dynamically impossible.

Proposed method

• Applying Kronecker's theorem on simultaneous Diophantine approximation to control the distribution of spectral points on the unit circle.
• Constructing a quasi-conjugacy between the operator $ T $ and a translation operator on the space of entire functions.
• Using the dynamics of iterates $ T^n $ to relate to translations $ f(\cdot + n) $ of an entire function $ f $, leveraging the spectral properties of $ T $.
• Analyzing the zero sets of real and imaginary parts of entire functions of exponential type to derive contradictions when spectral points are finite and linearly independent over $ \mathbb{Q} $.
• Employing the theory of functions of exponential type and density estimates to show that a product of such functions cannot vanish on a set of positive lower density unless identically zero.
• Deriving a contradiction by showing that the vanishing of certain entire functions implies they must be identically zero, violating the hypercyclicity assumption.

Experimental results

Research questions

• RQ1 Can a frequently hypercyclic operator have a spectrum contained in the closed unit disc with only finitely many points on the unit circle, provided the arguments of these points are linearly independent over $ \mathbb{Q} $?
• RQ2 Under what spectral conditions on the unit circle does frequent hypercyclicity become impossible for bounded Banach space operators?
• RQ3 Is it possible for a frequently hypercyclic operator to have a finite unimodular spectrum if the spectral arguments satisfy certain rational dependence or independence relations?
• RQ4 Can the spectral structure of an operator, particularly its intersection with the unit circle, be used to rule out frequent hypercyclicity via complex-analytic methods?
• RQ5 What role do entire functions of exponential type play in obstructing the existence of frequently hypercyclic operators with specific spectral configurations?

Key findings

• If the spectrum of a bounded operator $ T $ on a complex Banach space is contained in the closed unit disc and intersects the unit circle in a finite set $ \{e^{i\alpha_1}, \dots, e^{i\alpha_m}\} $, and if the $ \alpha_j $ are linearly independent over $ \mathbb{Q} $, then $ T $ cannot be frequently hypercyclic.
• The result extends Shkarin's theorem by showing that finite unimodular spectral sets with linearly independent arguments are incompatible with frequent hypercyclicity.
• If $ \alpha_1 n + r, \dots, \alpha_m n + r $ are linearly independent over $ \mathbb{Q} $ for some $ n \in \mathbb{N} $, $ r \in \mathbb{R} $, then $ T $ is not frequently hypercyclic.
• If $ \alpha_1/\pi, \dots, \alpha_m/\pi \in \mathbb{Q} $, then $ T $ is not frequently hypercyclic.
• For any finite set $ A = \sigma(T) \cap \mathbb{T} $ with at most two elements, $ T $ cannot be frequently hypercyclic.
• The contradiction arises from the fact that a product of $ 2m $ functions of exponential type less than $ d/(2t_0 m) $ cannot vanish on a set of lower density $ \geq d/t_0 $ unless identically zero, leading to a contradiction with the existence of a non-zero frequently hypercyclic vector.