The Cauchy dual subnormality problem asks whether the Cauchy dual operator of a 2-isometry is subnormal. Recently this problem has been solved in the negative. Here we show that it has a negative solution even in the class of cyclic 2-isometries.2010 Mathematics Subject Classification. Primary 47B20, 47B37 Secondary 44A60.

Let T be a rooted directed tree with finite branching index k T , and let S λ ∈ B(l 2 (V )) be a leftinvertible weighted shift on T . We show that S λ can be modelled as a multiplication operator Mz on a reproducing kernel Hilbert space H of E-valued holomorphic functions on a disc centred at the origin, where E := ker S * λ . The reproducing kernel associated with H is multi-diagonal and of bandwidth k T . Moreover, H admits an orthonormal basis consisting of polynomials in z with at most k T + 1 non-zero coefficients. As one of the applications of this model, we give a spectral picture of S λ . Unlike the case dim E = 1, the approximate point spectrum of S λ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed tree with finite branching index.Let H be a complex separable Hilbert space. The inner-product on H will be denoted by ·, · H . If no confusion is likely, then we suppress the suffix, and simply write the inner-product as ·, · . By a subspace, we mean a closed linear manifold. Let W be a subset of H. Then span W stands for the smallest linear manifold generated by W. In case W is singleton {w}, we use the convenient notation w in place of span {w}. By {w : w ∈ W }, we understand the subspace generated by W . For a subspace M of H, we use P M to denote the orthogonal projection of H onto M. For vectors x, y ∈ H, we use the notation x ⊗ y to denote the rank 1 operator given byUnless stated otherwise, all the Hilbert spaces occurring below are complex infinitedimensional separable and, for any such Hilbert space H, by B(H) we denote the Banach algebra of bounded linear operators on H. For T ∈ B(H), the symbols ker T and ran T will stand for the kernel and the range of T, respectively. The Hilbert space adjoint of T will be denoted by T * . In what follows, we denote the spectrum, approximate point spectrum, essential spectrum and the point spectrum of T by σ(T ), σ ap (T ), σ e (T ) and σ p (T ), respectively. We reserve the notation r(T ) for the spectral radius of T.Let T ∈ B(H). We say that T is left-invertible if there exists S ∈ B(H) such that ST = I. Note that T is left-invertible if and only if there exists a constant α > 0 such that T * T αI. In this case, T * T is invertible and T admits the left-inverse (T * T ) −1 T * . Following [30], we refer to the operator T given by T := T (T * T ) −1 as the Cauchy dual of the left-invertible operator T. Further, we say that T is analytic if n 0 T n (H) = {0}. If H is a reproducing kernel Hilbert space of holomorphic functions defined on a disc in C, then the multiplication operator M z defined on H provides an example of an analytic operator. It is interesting to note that almost all analytic operators arise in this way. Indeed, a result of Shimorin [30] asserts that any left-invertible analytic operator is unitarily equivalent to the operator of multiplication by z on a reproducing kernel Hilbert space of vector-valued holomorphic functions

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.

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