[Paper Review] On the chaoticity and spectral structure of Rolewicz-type unbounded operators

Marat V. Markin|arXiv (Cornell University)|Nov 16, 2018

Holomorphic and Operator Theory10 references4 citations

TL;DR

This paper establishes the chaoticity and complete spectral structure of Rolewicz-type unbounded weighted backward shift operators in $l_p$ ($1 \leq p < \infty$) and $c_0$ sequence spaces. It proves that for $|w| > 1$, these operators are hypercyclic and chaotic, with the entire complex plane as their spectrum, where every $\lambda \in \mathbb{C}$ is a simple eigenvalue with a one-dimensional eigenspace.

ABSTRACT

We prove the chaoticity and describe the spectral structure of Rolewicz-type weighted backward shift unbounded linear operators in the sequence spaces $l_p$ ($1\le p

Motivation & Objective

• To extend the theory of hypercyclicity and chaos from bounded to unbounded linear operators in Banach spaces.
• To analyze the dynamical behavior of unbounded weighted backward shift operators of Rolewicz type in $l_p$ and $c_0$ spaces.
• To characterize the spectral structure of these operators, particularly when the underlying space is complex.
• To demonstrate that every $\lambda \in \mathbb{C}$ is an eigenvalue with a one-dimensional eigenspace, leading to $\sigma(A) = \mathbb{C}$.

Proposed method

• Define the unbounded weighted backward shift operator $A$ on $X = l_p$ or $c_0$ via $Ax = (w^k x_{k+1})_{k \in \mathbb{N}}$, with domain $D(A) = \{x \in X : (w^k x_{k+1}) \in X\}$.
• Establish that $A$ is densely defined and closed by showing $C^\infty(A) \supset c_{00}$, the space of eventually zero sequences.
• Prove hypercyclicity by constructing a hypercyclic vector $x$ whose orbit under $A$ is dense in $X$, using a sequence with rapidly decaying components.
• Demonstrate chaos by showing the set of periodic points is dense, using sequences with periodic patterns in their coefficients.
• Analyze the eigenvalue equation $Ax = \lambda x$ to show that for each $\lambda \in \mathbb{C}$, the solution space is one-dimensional.
• Conclude that $\sigma(A) = \mathbb{C}$, with every $\lambda \in \mathbb{C}$ being an eigenvalue, by verifying the solution sequence lies in $D(A) \setminus \{0\}$.

Experimental results

Research questions

• RQ1 Is the Rolewicz-type unbounded weighted backward shift operator chaotic in $l_p$ and $c_0$ spaces for $|w| > 1$?
• RQ2 What is the spectral structure of such unbounded operators when the underlying space is complex?
• RQ3 Does every $\lambda \in \mathbb{C}$ become an eigenvalue of the operator, and if so, what is the dimension of the corresponding eigenspace?
• RQ4 Can the hypercyclicity and chaoticity of bounded Rolewicz operators be extended to their unbounded counterparts in sequence spaces?
• RQ5 How does the maximal domain $D(A)$ affect the dynamics and spectral properties of the operator?

Key findings

• The operator $A$ is chaotic in both $l_p$ ($1 \leq p < \infty$) and $c_0$ spaces for any $w \in \mathbb{F}$ with $|w| > 1$.
• The spectrum $\sigma(A)$ is the entire complex plane $\mathbb{C}$ when the underlying space is complex.
• Every $\lambda \in \mathbb{C}$ is an eigenvalue of $A$, with the corresponding eigenspace being one-dimensional.
• The eigenvector associated with $\lambda$ is given by $x_k = \left(\frac{\lambda}{w^{k/2}}\right)^{k-1} x_1$, which lies in $D(A) \setminus \{0\}$ for any $x_1 \in \mathbb{C}$.
• The set of periodic points of $A$ is dense in $X$, confirming the chaotic nature of the operator.
• The orbit of a carefully constructed hypercyclic vector is dense in $X$, proving that $A$ is hypercyclic.