Numerical Hypercyclicity of Composition Operators on Classical Function Spaces: Power Boundedness, Fixed Points, and Linear Dynamics
DOI:
https://doi.org/10.15377/2409-5761.2026.13.4Keywords:
Composition operators, Numerical hypercyclicity, Power bounded operators, Hardy and Bergman spaces, Hypercyclicity and linear dynamics.Abstract
This paper investigates numerical hypercyclicity for unweighted composition operators on several classical Banach and Hilbert spaces of functions. The aim is to compare this scalar orbit-density property with ordinary hypercyclicity and to determine when the two notions coincide. We consider composition operators on C([0,1]), H∞ (D), the disc algebra A(D), the Hardy space H2(D), the Bergman space A2(D), the Fock space F2(C), and the Dirichlet space D. The analysis combines power-boundedness arguments with known hypercyclicity criteria for composition operators. We show that power boundedness rules out both hypercyclicity and numerical hypercyclicity on C([0,1]), H∞ (D), A(D), and for bounded composition operators on F2(C). In contrast, for automorphism-induced composition operators on H2(D) and A2(D), numerical hypercyclicity is equivalent to ordinary hypercyclicity, and hence to the absence of fixed points of the symbol in D. For the Dirichlet space, we obtain partial results: elliptic automorphisms are shown not to induce numerically hypercyclic composition operators, while the remaining automorphic cases are reduced to an explicit scalar-orbit problem. The novelty of the paper lies in providing a unified and space-dependent comparison between numerical hypercyclicity and hypercyclicity for composition operators across these classical settings.
2020 Mathematics Subject Classification: 47A16, 47B33, 30H10, 47B38.
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Copyright (c) 2026 Hadi O. Alshammari, Otmane Benchiheb, Marko Kostić
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