Weakly concave operators

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 A(1), in the chain A(0) subset of A(1) subset of... subset of A(infinity) 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 A(k) is finite or not. In particular, a Berger-Shaw-type theorem fails to be true for members of A(infinity). This discrepancy is better revealed in the context of C*- and W*-algebras.