[Paper Review] On general construct of chaotic unbounded linear operators in Banach Spaces with Schauder Bases

Marat V. Markin|arXiv (Cornell University)|Dec 6, 2018

Holomorphic and Operator Theory9 references4 citations

TL;DR

This paper presents a general construction of chaotic unbounded linear operators in Banach spaces with Schauder bases, extending Rolewicz's weighted backward shift to infinite-dimensional settings. By defining a weighted backward shift operator with exponential weights satisfying specific decay conditions, the authors prove the operator is both hypercyclic and chaotic, generalizing prior results on $ l_p $ and $ c_0 $ spaces to arbitrary Banach spaces with Schauder bases.

ABSTRACT

We utilize the idea underlying the construct of the classical weighted backward shift Rolewicz's operators to furnish a straightforward approach to a general construct of chaotic unbounded linear operators in a (real or complex) Banach space with a Schauder basis.

Motivation & Objective

• To extend the theory of hypercyclicity and chaoticity from bounded to unbounded linear operators in Banach spaces.
• To generalize Rolewicz's classical weighted backward shift example to unbounded operators in general Banach spaces with Schauder bases.
• To establish sufficient conditions under which such unbounded operators are both hypercyclic and chaotic.
• To prove that the set of periodic points is dense in the space, confirming chaoticity.
• To unify and generalize existing results on hypercyclic and chaotic operators in $ l_p $ and $ c_0 $ spaces.

Proposed method

• Define a weighted backward shift operator $ A $ on a Banach space $ X $ with a Schauder basis using a weight sequence $ \{w_n\} $ satisfying $ |w_n| > 1 $ and $ \sum_{n=1}^\infty |w_n|^{-1} < \infty $.
• Construct the operator via $ Ax = (w_k x_{k+1})_{k\in\mathbb{N}} $ with domain $ D(A) = \{ x \in X \mid (w_k x_{k+1}) \in X \} $.
• Use the convergence of the Schauder expansion and the decay of $ |w_n|^{-1} $ to ensure the orbit of a carefully chosen vector is dense in $ X $.
• Prove hypercyclicity by showing the orbit of a specific vector $ x \in C^\infty(A) $ is dense using estimates on the tail of the series.
• Demonstrate chaoticity by constructing periodic points of arbitrary period $ N $, and showing their set is dense in $ X $ via approximation from a countable dense subset $ Y $.
• Leverage the closedness of $ A^n $ and the structure of the Schauder basis to ensure $ x \in D(A^n) $ and $ A^n x \to x $ in norm.

Experimental results

Research questions

• RQ1 Can the classical construction of hypercyclic weighted backward shifts be generalized to unbounded linear operators in arbitrary Banach spaces with Schauder bases?
• RQ2 What conditions on the weight sequence ensure hypercyclicity and chaoticity for unbounded operators in such spaces?
• RQ3 Is it possible to construct a chaotic unbounded linear operator in a Banach space with a Schauder basis that is not isomorphic to $ l_p $ or $ c_0 $?
• RQ4 How can the denseness of periodic points be established for unbounded operators in this general setting?
• RQ5 Does the exponential weight sequence $ w_n = \lambda^n $ with $ |\lambda| > 1 $ yield a chaotic unbounded operator in any Banach space with a Schauder basis?

Key findings

• The constructed unbounded weighted backward shift operator is hypercyclic in any Banach space with a Schauder basis, provided the weight sequence satisfies $ \sum_{n=1}^\infty |w_n|^{-1} < \infty $ and $ |w_n| > 1 $.
• The operator is chaotic, as the set of periodic points is dense in the Banach space, which is established by approximating arbitrary vectors in a countable dense subset $ Y $ with periodic points of increasing period.
• For the exponential weight $ w_n = \lambda^n $ with $ |\lambda| > 1 $, the operator $ Ax = (\lambda^k x_{k+1})_{k\in\mathbb{N}} $ is chaotic on $ l_p $ ( $ 1 \leq p < \infty $) and $ c_0 $, recovering and strengthening Theorem 1.1 from prior work.
• The vector $ x \in C^\infty(A) $ constructed via the Schauder basis and weight sequence ensures the orbit $ \{A^n x\}_{n\in\mathbb{Z}_+} $ is dense in $ X $, proving hypercyclicity.
• The proof relies on estimating the tail of the series $ \sum_{j=m+1}^\infty B^{n_j - n_m} y^{(j)} $, showing it converges and tends to zero as $ m \to \infty $, which implies orbit denseness.
• The construction is general and applies to all separable Banach spaces with Schauder bases, including $ L_p(a,b) $, $ C[a,b] $, and $ l_p $, $ c_0 $, $ c $, demonstrating broad applicability.