On compressions and generalized spectra of operators over C*-algebras

TL;DR

This paper extends classical spectral concepts to adjointable operators on Hilbert C*-modules, demonstrating that many spectral relations hold in this more general setting, including those involving operator compressions.

Contribution

It introduces generalized spectra for operators on Hilbert C*-modules and establishes that classical spectral relations persist in this broader context.

Findings

Most spectral relations are preserved in the generalized setting.

Relations between spectra of operators and their compressions are extended to Hilbert C*-modules.

The generalized spectra are related to the center of the C*-algebra.

Abstract

In the classical operator theory, there are several versions of spectra, related to special classes of operators (Fredholm, semi-Fredholm, upper/lower semi-Fredholm,etc.). We generalize these notions for adjointable operators on Hilbert C*-modules replacing scalars by the center of the algebra, and show that most relations between these spectra are still true for these generalized versions. The relation between these spectra of an operator and those of its compressions is also transferred to the case of Hilbert C*-modules.