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

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 Cd 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 HB(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(C2)-valued kernels for which the associated multiplication operators Mz are non-hyponormal reflexive operators with commutants equal to HB(C2)∞(D).