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We systematically develop the multivariable counterpart of the theory ofweighted shifts on rooted directed trees. Capitalizing on the theory of productof directed graphs, we introduce and study the notion of multishifts ondirected Cartesian product of rooted directed trees. This framework unifies thetheory of weighted shifts on rooted directed trees and that of classicalunilateral multishifts. Moreover, this setup brings into picture some newphenomena such as the appearance of system of linear equations in theeigenvalue problem for the adjoint of a multishift. In the first half of thepaper, we focus our attention mostly on the multivariable spectral theory andfunction theory including finer analysis of various joint spectra and wanderingsubspace property for multishifts. In the second half, we separate out twospecial classes of multishifts, which we refer to as torally balanced andspherically balanced multishifts. The classification of these two classes isclosely related to toral and spherical polar decompositions of multishifts.Furthermore, we exhibit a family of spherically balanced multishifts on$d$-fold directed Cartesian product $\mathscr T$ of rooted directed trees.These multishifts turn out be multiplication $d$-tuples $\mathscr M_{z, a}$ oncertain reproducing kernel Hilbert spaces $\mathscr H_a$ of vector-valuedholomorphic 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 actingon the reproducing kernel Hilbert spaces associated with the kernels$\frac{1}{(1-\langle{z},{{w}\rangle})^a}~(z, w \in \mathbb B^d, a \in \mathbbN).$

Multishifts on Directed Cartesian Product of Rooted Directed Trees