[Paper Review] Hypercyclic subspaces and weighted shifts

Quentin Menet|arXiv (Cornell University)|Aug 24, 2012

Holomorphic and Operator Theory10 references7 citations

TL;DR

This paper generalizes hypercyclic subspace criteria to sequences of operators on Fréchet spaces with a continuous norm and establishes a simple criterion for the absence of hypercyclic subspaces in weighted shifts. It characterizes weighted shifts on $\ell^p$, $c_0$, the space of entire functions, and certain Köthe sequence spaces that possess hypercyclic subspaces, proving that every non-constant polynomial of the differentiation operator on entire functions has a hypercyclic subspace.

ABSTRACT

We first generalize the results of León and Müller [Studia Math. 175(1) 2006] on hypercyclic subspaces to sequences of operators on Fréchet spaces with a continuous norm. Then we study the particular case of iterates of an operator T and show a simple criterion for having no hypercyclic subspace. Finally we deduce from this criterion a characterization of weighted shifts with hypercyclic subspaces on the spaces lp or c0, on the space of entire functions and on certain Köthe sequence spaces. We also prove that if P is a non-constant polynomial and D is the differentiation operator on the space of entire functions then P(D) possesses a hypercyclic subspace.

Motivation & Objective

• To extend the theory of hypercyclic subspaces from Banach spaces to Fréchet spaces with a continuous norm.
• To develop a simple criterion for when an operator on a Banach or Fréchet space lacks a hypercyclic subspace.
• To characterize weighted shifts on $\ell^p$, $c_0$, the space of entire functions, and certain Köthe sequence spaces that admit hypercyclic subspaces.
• To prove that every non-constant polynomial of the differentiation operator on the space of entire functions has a hypercyclic subspace.

Proposed method

• Generalizing León and Müller’s criteria (Theorems 0.3 and 0.4) to Fréchet spaces with a continuous norm.
• Introducing a new criterion based on uniform lower bounds on operator norms over finite-codimensional subspaces to detect the absence of hypercyclic subspaces.
• Applying the criterion to weighted shifts by analyzing growth conditions on weights and associated sequence space norms.
• Using matrix representations and asymptotic comparisons of weight sequences to verify the hypercyclicity criterion in Köthe sequence spaces.
• Leveraging spectral and functional analytic tools, including the Hypercyclicity Criterion and properties of the differentiation operator on $H(\mathbb{C})$.
• Establishing sufficient conditions involving limsup and liminf comparisons of weight and sequence space coefficients to guarantee the existence of hypercyclic subspaces.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of operators on a Fréchet space with a continuous norm possess a hypercyclic subspace?
• RQ2 When does an operator on a Banach or Fréchet space fail to have a hypercyclic subspace?
• RQ3 Which weighted shifts on $\ell^p$, $c_0$, or the space of entire functions possess hypercyclic subspaces?
• RQ4 Does every non-constant polynomial of the differentiation operator on the space of entire functions have a hypercyclic subspace?
• RQ5 What are the necessary and sufficient conditions for a weighted shift on a Köthe sequence space to admit a hypercyclic subspace?

Key findings

• The criterion in Theorem 0.5 provides an equivalent condition for the absence of a hypercyclic subspace in terms of uniform lower bounds on $\|T^n x\|$ over finite-codimensional subspaces.
• For weighted shifts on $\ell^p$ or $c_0$, if $\inf_k \prod_{\nu=1}^n |w_{\nu+k}| = 0$ for all $n \geq 1$, then $I + B_w$ has a hypercyclic subspace.
• On the space of entire functions, every non-constant polynomial $P(D)$ of the differentiation operator $D$ possesses a hypercyclic subspace.
• In Köthe sequence spaces $\lambda^p(A)$ or $c_0(A)$, a weighted shift $B_w$ has a hypercyclic subspace if certain limsup and liminf conditions on weight and sequence coefficients are satisfied.
• If $\lim_k \frac{a_{j,k} a_{m,n+k}}{a_{J,k} a_{m_j,n+k}} = 0$ uniformly and $\inf_k \frac{\prod_{\nu=1}^n |w_{\nu+k}| a_{J,k}}{a_{m,n+k}} = 0$, then $P(B_w)$ has a hypercyclic subspace provided $P(B_w)$ satisfies the Hypercyclicity Criterion.
• The space of entire functions and the space of rapidly decreasing sequences satisfy the required growth conditions for the existence of hypercyclic subspaces in weighted shifts.