On the dynamics of Toeplitz operators over Bergman spaces
Abstract
We investigate the hypercyclicity of Toeplitz operators on the Bergman space LA2(𝔻) with symbols of the form Ψ(z) = γz̅+ψ(z), where γ∈ ℂ ∖ {0} and ψ is analytic on an open neighborhood of the closed unit disc 𝔻̅. Our approach bypasses the classical Hardy space techniques (reproducing kernel linear combinations, Nevanlinna factorization) by directly solving the integro-differential resolvent equation arising from the Bergman projection. A key winding number argument shows that for sense-reversing symbols (|γ| > supz ∈ 𝔻̅ |ψ'(z)|), the symbol's image Ψ(𝔻) is contained in the point spectrum of TΨ. In the tridiagonal case Ψ(z) = az̅+b+cz, we fully resolve the longstanding eigenvector completeness problem by linking the recurrence coefficients to a rotated Favard spectral measure on the major axis of the symbol's ellipse. Combined with self-commutator positivity, this establishes an unconditional, exact necessary and sufficient characterization of hypercyclicity: TΨ is hypercyclic if and only if |a| > |c| and Ψ(𝔻) intersects both the unit disc and its exterior, completely eliminating the (3+√2) restriction of previous literature.
Related
- Toeplitz and Hankel operators on Bergman spaces with doubling weights
- Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
- Generalized Volterra Companion Operators on Bergman Spaces over Convex Domains of Finite Type
- Absolutely summing operators on Bergman spaces over convex domains of finite type
- Commuting analytic and co-analytic Toeplitz operators on the harmonic Bergman space
BibTeX
@misc{abad2026dynamics,
title = {{On the dynamics of Toeplitz operators over Bergman spaces}},
author = {Othman Abad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22017},
eprint = {2609.22017},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22017},
}
RIS
TY - UNPB
TI - On the dynamics of Toeplitz operators over Bergman spaces
AU - Othman Abad
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22017
UR - https://arxiv.org/abs/2609.22017
AB - We investigate the hypercyclicity of Toeplitz operators on the Bergman space L_A^2(𝔻) with symbols of the form Ψ(z) = γz̅+ψ(z), where γ∈ ℂ ∖ {0} and ψ is analytic on an open neighborhood of the closed unit disc 𝔻̅. Our approach bypasses the classical Hardy space techniques (reproducing kernel linear combinations, Nevanlinna factorization) by directly solving the integro-differential resolvent equation arising from the Bergman projection. A key winding number argument shows that for sense-reversing symbols (|γ| > sup_z ∈ 𝔻̅ |ψ'(z)|), the symbol's image Ψ(𝔻) is contained in the point spectrum of T_Ψ. In the tridiagonal case Ψ(z) = az̅+b+cz, we fully resolve the longstanding eigenvector completeness problem by linking the recurrence coefficients to a rotated Favard spectral measure on the major axis of the symbol's ellipse. Combined with self-commutator positivity, this establishes an unconditional, exact necessary and sufficient characterization of hypercyclicity: T_Ψ is hypercyclic if and only if |a| > |c| and Ψ(𝔻) intersects both the unit disc and its exterior, completely eliminating the (3+√2) restriction of previous literature.
ER -