Perturbations of Truncated Toeplitz operators
Abstract
We investigate the structure of kernels of finite-rank perturbations of truncated Toeplitz operators. By realizing a perturbed truncated Toeplitz operator on a model space as a suitable matrix-valued truncated Toeplitz operator on an enlarged model space, we identify its kernel with a coordinate projection of the kernel of an associated block Toeplitz operator. This realization enables us to transfer the nearly S*-invariant structure of Toeplitz kernels to the perturbed truncated setting. More precisely, for a perturbation of rank at most n, we show that the kernel of the perturbed truncated Toeplitz operator is nearly S*-invariant with defect at most 2n+1. We then extend the method to truncated Toeplitz operators on multiband spaces. In this setting, a perturbation of rank at most n again preserves the finite-defect nearly S*-invariant structure of the kernel, with the explicit upper bound 2n+2 for the defect.
Related
- Block Toeplitz Operators, Matrix Valued Truncated Toeplitz operators, and Conjugations
- Algebraic properties of Truncated Toeplitz + Truncated Hankel operators
- Matrix-Valued Truncated Toeplitz Operators: Properties and Applications
- Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices
- Characterizations of Toeplitz operators on the annulus
BibTeX
@misc{chattopadhyay2026perturbations,
title = {{Perturbations of Truncated Toeplitz operators}},
author = {Arup Chattopadhyay and Arshad Khan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40295},
eprint = {2610.40295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Perturbations of Truncated Toeplitz operators
AU - Arup Chattopadhyay
AU - Arshad Khan
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40295
UR - https://arxiv.org/abs/2610.40295
AB - We investigate the structure of kernels of finite-rank perturbations of truncated Toeplitz operators. By realizing a perturbed truncated Toeplitz operator on a model space as a suitable matrix-valued truncated Toeplitz operator on an enlarged model space, we identify its kernel with a coordinate projection of the kernel of an associated block Toeplitz operator. This realization enables us to transfer the nearly S^*-invariant structure of Toeplitz kernels to the perturbed truncated setting. More precisely, for a perturbation of rank at most n, we show that the kernel of the perturbed truncated Toeplitz operator is nearly S^*-invariant with defect at most 2n+1. We then extend the method to truncated Toeplitz operators on multiband spaces. In this setting, a perturbation of rank at most n again preserves the finite-defect nearly S^*-invariant structure of the kernel, with the explicit upper bound 2n+2 for the defect.
ER -