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 $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $\Omega$, where $E$ is a separable Hilbert space and $\Omega$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the commutant of $\mathscr M_z$ is shown to be the algebra $H^{\infty}_{_{B(E)}}(\Omega)$ of bounded holomorphic $B(E)$-valued functions on $\Omega$, provided $\mathscr M_z$ 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 $\mathscr T$ of finite branching index. As the second main result of this paper, we show that a multiplication $d$-tuple $\mathscr M_z$ on $\mathscr H$ 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 $B(\mathbb C^2)$-valued kernels for which the associated multiplication operators $\mathscr M_z$ are non-hyponormal reflexive operators with commutants equal to $H^{\infty}_{_{B(\mathbb C^2)}}(\mathbb D)$.