Equality of uniform and Carleman spectra for bounded measurable functions

TL;DR

This paper proves the equivalence of uniform, Carleman, and Laplace spectra for bounded measurable functions in Banach spaces, extending spectral analysis tools in functional analysis.

Contribution

It establishes the equality of different spectral notions for bounded measurable functions, including new results for Laplace spectra.

Findings

Uniform spectrum coincides with Carleman spectrum for bounded functions.

Laplace spectrum equals Carleman spectrum for certain classes of functions.

Results extend spectral analysis techniques in Banach space theory.

Abstract

In this paper we study various types of spectra of functions $\phi:\jj\to X$, where $\jj\in\{\r_+,\r\}$ and $X$ is a complex Banach space. We show that uniform spectrum defined in [15] coincides with Carleman spectrum for $\phi\in L^{\infty}(\r,X)$. This result holds true also for Laplace (half-line) spectrum for $\phi\in L^{\infty}(\r_+,X)$. We also indicate a class of bounded measurable functions for which Laplace spectrum and Carleman spectrum are equal