  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
$H^{\infty}_{_{B(\mathbb C^2)}}(\mathbb D)$.
