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.
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