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 pathways to new ideas will never close; the doors of wisdom (literature) will always be open, till the very end. 1

The wandering subspace problem for an analytic normincreasing m-isometry T on a Hilbert space H asks whether every Tinvariant subspace of H can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 is ascribed to Beurling-Lax-Halmos, while that for m = 2 is due to Richter. In this paper, we capitalize on the idea of weighted shift on one-circuit directed graph to construct a family of analytic cyclic 3-isometries, which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one dimensional space, their norms can be made arbitrarily close to 1. We also show that if the wandering subspace property fails for an analytic norm-increasing m-isometry, then it fails miserably in the sense that the smallest T -invariant subspace generated by the wandering subspace is of infinite codimension.

Abstract. Let T = (V, E) be a leafless, locally finite rooted directed tree. We associate with T a one parameter family of Dirichlet spaces Hq (q 1), which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc D in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernelwhere V≺ denotes the set of branching vertices of T , nv denotes the depth of v ∈ V in T , and P eroot , Pv (v ∈ V≺) are certain orthogonal projections.Further, we discuss the question of unitary equivalence of operators M(1) z and M (2) z of multiplication by z on Dirichlet spaces Hq associated with directed trees T 1 and T 2 respectively.

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication d-tuple Mz on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in C d admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for Mz, under some natural conditions on the B(E)-valued kernel associated with H , the commutant of Mz is shown to be the algebra H ∞ B(E) (Ω) of bounded holomorphic B(E)-valued functions on Ω, provided Mz satisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of dim E = 1 and d = 1). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed tree T of finite branching index. As the second main result of this paper, we show that a multiplication d-tuple Mz on H satisfying the von Neumann's inequality is reflexive. This provides several new classes of examples as well as recovers special cases of various known results in one and several variables. We also exhibit a family of tri-diagonal B(C 2 )-valued kernels for which the associated multiplication operators Mz are non-hyponormal reflexive operators with commutants equal to H ∞ B(C 2 ) (D).2010 Mathematics Subject Classification. Primary 46E22, 47A13, Secondary 46E40, 47B37.