Multishifts on directed Cartesian products of rooted directed trees
Volume 527 / 2017
Abstract
We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of products of directed graphs, we introduce and study the notion of multishifts on directed Cartesian products of rooted directed trees. This framework unifies the theory of weighted shifts on rooted directed trees and that of classical unilateral multishifts. Moreover, this setup brings out some new phenomena such as the appearance of a system of linear equations in the eigenvalue problem for the adjoint of a multishift. In the first half of the paper, we focus our attention mostly on the multivariable spectral theory and function theory including a finer analysis of various joint spectra and the wandering subspace property for multishifts. In the second half, we separate out two special classes of multishifts, which we refer to as torally balanced and spherically balanced multishifts. The classification of these two classes is closely related to toral and spherical polar decompositions of multishifts. Furthermore, we exhibit a family of spherically balanced multishifts on a $d$-fold directed Cartesian product $\mathscr T$ of rooted directed trees. These multishifts turn out to be multiplication $d$-tuples $\mathscr M_{z, a}$ on certain reproducing kernel Hilbert spaces $\mathscr H_a$ of vector-valued holomorphic functions defined on the unit ball $\mathbb B^d$ in $\mathbb C^d$, which can be thought of as tree analogs of the multiplication $d$-tuples acting on the reproducing kernel Hilbert spaces associated with the kernels $\newcommand{\inp}[2]{\langle{#1},{#2} \rangle}{1}/{(1-\inp{z}{{w}})^a} (z, w \in \mathbb B^d, a \in \mathbb N).$ Indeed, the reproducing kernels associated with $\mathscr H_a$ are certain operator linear combinations of $\newcommand{\inp}[2]{\langle{#1},{#2} \rangle}{1}/{(1-\inp{z}{{w}})^a}$ and multivariable hypergeometric functions ${}_2F_1(\mathsf{d}_v+a+1, 1, \mathsf{d}_v+2, \cdot)$ defined on $\mathbb B^d \times \mathbb B^d$, where $\mathsf{d}_v$ denotes the depth of a branching vertex $v$ in $\mathscr T$. We also classify joint subnormal and joint hyponormal multishifts within the class of spherically balanced multishifts.
