Let $\mathbb B_d$ be the open unit ball in $\mathbb C^d$ and $\boldsymbol T$ be a commuting $d$-tuple of bounded linear operators on a complex separable Hilbert space $\mathcal H$. Let $\mathcal U(d)$ be the linear group of unitary transformations acting on $\mathbb C^d$ by the rule: $\boldsymbol z \mapsto u\cdot \boldsymbol z$, $\boldsymbol z \in \mathbb C^d$, where $u\cdot \boldsymbol z$ is the usual matrix product. We say that $\boldsymbol T$ is $\mathcal U(d)$-homogeneous if $u \cdot \boldsymbol T$ is unitarily equivalent to $\boldsymbol T$ for all $u\in \mathcal U(d)$. In this talk, we describe $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol M$ of multiplication by the coordinate functions acting on a reproducing kernel Hilbert space $\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq {\rm Hol}(\mathbb B_d, \mathbb C^n)$, where $n$ is the dimension of the joint kernel of $\boldsymbol T^*$. The case $n=1$ is well understood, here, we focus on the case $n=d.$ We describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative definite kernels quasi invariant under the action of the group $\mathcal U(d).$ As a result, we obtain criterion for boundedness, irreducibility and mutual unitary equivalence among these operators.

This is a joint work with Soumitra Ghara, Gadadhar Misra and Paramita Pramanick.