2021/08/23 by Sameer Chavan, Jan Stochel, Chavan, Sameer +1 · 1 citation
Mathematics · #47A10 #47C15 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 47B20 #Secondary 47A16 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2108.10228
openalex publication_date 2021/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A1, in the chain \mathcal A0 ⊆ \mathcal A1 ⊆ … ⊆ \mathcal A∞ 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 Ak is finite or not. In particular, a Berger-Shaw-type theorem fails to be true for members of~\mathcal A∞. This discrepancy is better revealed in the context of C^*- and W^*-algebras.