Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

Abstract

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.

Generalized multipliers for left-invertible analytic operators and their applications to commutant and reflexivity

“…First, we are going to prove thatφ n (m) SOT −→φ(m) for every m ∈ N 0 . Fix e ∈ E and let g = U e. Then by Lemma 1 (i), (15), (6) and (13) we obtain ϕ n (m)e = Mφ n g(m)…”

“…Proof. (i) Fix e ∈ E and let f = U e. Then by (9), (6), and equality N (L) = E we get φ * U e (n) = n k=0φ (k)P E L n−k e =φ(n)e, n ∈ N 0 .…”

“…It is a large class of operators. The examples can be found in [6,Section 5] or in Example 34. First, let us note that the orthogonality of two vectors concentrated on one generation (i.e.…”

“…. , z n on a reproducing Hilbert spaces of E-valued holomorphic on bounded domain in C n admitting polynomial approximation, where E is a separable Hilbert space, were studied in the paper [6]. In particular, some new results for commutant of Bergmann weighted shifts (which are automatically balanced) on rooted, locally finite, leafless directed trees with finite branching index (with finite dimensional kernel of the adjoint) were proven.…”

New Numerical Method for Solving Tenth Order Boundary Value Problems

“…Aronszajn has given a systematic reproducing kernel theory containing the Bergman kernel function [10]. For more details, see [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”