Weakly concave operators

TL;DR

This paper introduces weakly concave operators, a class of left-invertible operators, and establishes spectral and structural theorems including a Wold-type decomposition and Berger-Shaw-type results, extending to related classes with nuanced differences.

Contribution

It defines weakly concave operators and proves key spectral and structural theorems, extending classical results to this new class and related subclasses of left-invertible operators.

Findings

Wold-type decomposition for weakly concave operators

Berger-Shaw-type theorem for finitely cyclic weakly concave operators

Spectral dichotomy relating spectra of operators and their inverses

Abstract

We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$-isometries with $m > 2$. We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger-Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $\mathcal A_1$, in the chain $\mathcal A_0 \subseteq \mathcal A_1 \subseteq \ldots \subseteq \mathcal A_{\infty}$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $\mathcal A_k$ is finite or not. In particular, a Berger-Shaw-type theorem fails to be true for members of~$\mathcal A_{\infty}$. This discrepancy is better revealed in the context of $C^*$- and $W^*$-algebras.