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.

The operator Cauchy dual to a $2$-hyperexpansive operator $T$, given by $T'\equiv T(T^*T)^{-1}$, turns out to be a hyponormal contraction. This simple observation leads to a structure theorem for the $C^*$-algebra generated by a $2$-hyperexpansion, and a version of the Berger–Shaw theorem for $2$-hyperexpansions.

As an application of the hyperexpansivity version of the Berger–Shaw theorem, we show that every analytic $2$-hyperexpansive operator with finite-dimensional cokernel is unitarily equivalent to a compact perturbation of a unilateral shift.