Weakly concave operators

We study a class of left-invertible operators which we call weakly concave operators. It includes the and some subclasses expansive strict $m$ -isometries with $m > 2$ . prove Wold-type decomposition for also obtain Berger–Shaw-type theorem analytic finitely cyclic The proofs these results rely heavily on spectral dichotomy provides fairly close relationship, written in terms reciprocal automorphism Riemann sphere, between spectra operator any its left inverses. further place operators, as term $\mathcal {A}_1$ , chain {A}_0 \subseteq \mathcal {A}_1 \ldots {A}_{\infty }$ collections show that most aforementioned can be proved members classes. Subtleties arise depending whether index $k$ {A}_k$ is finite or not. In particular, fails to true This discrepancy better revealed context $C^*$ - $W^*$ -algebras.