Spectral Properties of Convolution Operators on Non-Separable Sequence Spaces
Abstract
This paper provides a complete characterization of the spectrum, point spectrum, continuous spectrum, and residual spectrum of bounded convolution operators acting on the non-separable space σ∞, the dual of the Hahn sequence space. Because standard duality and separability-based techniques fail in this setting, a linear functional approach is employed with the fine spectral decomposition of the unilateral shift operator. Furthermore, as a major application of this methodology, we establish a complete description of the fine spectrum for operator polynomials acting on general sequence spaces without requiring separability. This result provides a unified framework that subsumes classical spectral determinations for general banded triangular matrices across various sequence spaces and enables the explicit computation of fine spectrum on non-separable spaces.
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BibTeX
@misc{saha2026spectral,
title = {{Spectral Properties of Convolution Operators on Non-Separable Sequence Spaces}},
author = {Sayan Saha and Arnab Patra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02919},
eprint = {2610.02919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02919},
}
RIS
TY - UNPB
TI - Spectral Properties of Convolution Operators on Non-Separable Sequence Spaces
AU - Sayan Saha
AU - Arnab Patra
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02919
UR - https://arxiv.org/abs/2610.02919
AB - This paper provides a complete characterization of the spectrum, point spectrum, continuous spectrum, and residual spectrum of bounded convolution operators acting on the non-separable space σ_∞, the dual of the Hahn sequence space. Because standard duality and separability-based techniques fail in this setting, a linear functional approach is employed with the fine spectral decomposition of the unilateral shift operator. Furthermore, as a major application of this methodology, we establish a complete description of the fine spectrum for operator polynomials acting on general sequence spaces without requiring separability. This result provides a unified framework that subsumes classical spectral determinations for general banded triangular matrices across various sequence spaces and enables the explicit computation of fine spectrum on non-separable spaces.
ER -