Equality of uniform and Carleman spectra for bounded measurable functions

Abstract

In this paper we study various types of spectra of functions φ : J → X, where J ∈ {R + , R} and X is a complex Banach space. We show that uniform spectrum defined in [15] coincides with Carleman spectrum for φ ∈ L ∞ (R, X). This result holds true also for Laplace (half-line) spectrum for φ ∈ L ∞ (R + , X). We also indicate a class of bounded measurable functions for which Laplace spectrum and Carleman spectrum are equal.