Spectral cuts and unconventional functional calculi

TL;DR

This paper explores operators on Banach and Hilbert spaces that admit spectral cuts and unconventional functional calculi, revealing new classes of such operators and their properties, including invariant subspaces and super-decomposability.

Contribution

It introduces new classes of operators with spectral cuts that admit unconventional functional calculi and extends previous results on invariant subspaces and super-decomposability.

Findings

Operators with spectral cuts along Jordan curves are linked to unconventional functional calculi.

Trace-class perturbations of diagonalizable normal operators are super-decomposable.

Extended the class of operators known to admit unconventional functional calculus.

Abstract

In this work, we prove that linear bounded operators $T$ on a Banach space $X$ allowing spectral cuts along rectifiable Jordan curves meeting their spectrum are related to classes of operators admitting an unconventional functional calculus. We identify several such classes and address the consequences regarding the existence of non-trivial closed invariant subspaces, extending previous results of Chalendar. Furthermore, we establish that every operator belonging to a broad subclass of compact perturbations of diagonalizable normal operators on separable Hilbert spaces, namely, trace-class perturbations, possesses an unconventional functional calculus and is super-decomposable, thereby extending earlier results obtained by the authors.