Multishifts on Directed Cartesian Product of Rooted Directed Trees

Abstract

We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of product of directed graphs, we introduce and study the notion of multishifts on directed Cartesian product of rooted directed trees. This framework uni es the theory of weighted shifts on rooted directed trees and that of classical unilateral multishifts. Moreover, this setup brings into picture some new phenomena such as the appearance of system of linear equations in the eigenvalue problem for the adjoint of a multishift. In the rst half of the paper, we focus our attention mostly on the multivariable spectral theory and function theory including ner analysis of various joint spectra and 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 classi cation 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 d-fold directed Cartesian product T of rooted directed trees. These multishifts turn out be multiplication d-tuples Mz;a on certain reproducing kernel Hilbert spaces Ha of vector-valued holomorphic functions de ned on the unit ball Bd in Cd, which can be thought of as tree analogs of the multiplication d-tuples acting on the reproducing kernel Hilbert spaces associated with the kernels 1 (1􀀀hz;wi)a (z;w 2 Bd; a 2 N): Indeed, the reproducing kernels associated with Ha are certain operator linear combinations of 1 (1􀀀hz;wi)a and multivariable hypergeometric functions 2F1( v + a + 1; 1; v + 2; ) de ned on Bd Bd, where v denotes the depth of a branching vertex v in T . We also classify joint subnormal and joint hyponormal multishifts within the class of spherically balanced multishifts.