Linear dynamics of the adjoint of a unilateral weighted shift operator

2024/12/07 by Bibhash Kumar Das, Das, Bibhash Kumar, Aneesh Mundayadan +1
Mathematics · #47A16 #Advanced Banach Space Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2412.05509

openalex publication_date 2024/12/07 · openalex created_date 2024/12/12 · openalex updated_date 2026/07/28

Abstract

This paper is a sequel to our work in \citeDas-Mundayadan. Here, we primarily study the dynamics of the adjoint of a weighted forward shift operator F_w on the analytic function space ℓ^p_a,b having a normalized Schauder basis of the form \(a_n+b_nz)zⁿ:~n ≥ 0\. We obtain sufficient conditions for F_w to be continuous, and show, under certain conditions, that the operator F_w is similar to a compact perturbation of a weighted forward shift on ℓ^p(ℕ₀). This also allows us to obtain the essential spectrum of F_w. Further, we study when the adjoint F_w^* is hypercyclic, mixing, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on ℓ^p(ℕ₀). Finally, it is proved that the adjoint of a shift on the dual of ℓ^p_a,b can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero-one law of orbital limit points fails for F_w^*, which means that, under certain conditions, the adjoint F_w^* is non-hypercyclic, but it has an orbit possessing non-zero norm limit points.

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