非可分序列空间上卷积算子的谱性质

中文摘要 AI 辅助

本文给出了作用在非可分空间$\sigma_\infty$（Hahn序列空间的对偶空间）上的有界卷积算子的谱、点谱、连续谱和剩余谱的完整刻画。由于标准对偶性和基于可分性的技术在此背景下失效，我们采用线性泛函方法，并结合单侧移位算子的精细谱分解。此外，作为该方法的一个主要应用，我们在不要求可分性的情况下，建立了作用在一般序列空间上的算子多项式的精细谱的完整描述。这一结果提供了一个统一框架，涵盖了各种序列空间上一般带状三角矩阵的经典谱确定，并使得在非可分空间上能够显式计算精细谱。

英文摘要

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 $σ_\infty$, 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.

发表机构

• Indian Institute of Technology Bhilai（印度理工学院比莱分校）

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