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 star-shaped domain in $\mathbb C^d$ with polynomially convex closure. 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 leafl 查看全文>>