Complete Nevanlinna-Pick property of $\mathbb{K}$-Invariant Reproducing Kernels

Let $\mathrm{\Omega }$ be a Cartan domain and $K={\sum }_{\underset{‾}{s}}{a}_{\underset{‾}{s}}{K}_{\underset{‾}{s}}$ be a $\mathbb{K}$-invariant kernel on $\mathrm{\Omega }$. In this article, we first obtain a necessary condition on $K$ to have the complete Nevanlinna-Pick property in terms of the sequence $\{a_{\underset{‾}{s}}{\}}_{\underset{‾}{s}}$ with the assumption that each $a_{\underset{‾}{s}}$ is non-zero and $K$ is non-vanishing. This generalizes the well-known Kaluza’s Lemma in the context of $\mathbb{K}$-invariant kernels. The notion of the characteristic function of the classical Sz.-Nagy–Foias Theory is extended to a commuting tuple of $\frac{1}{K}$-contraction where $K$ is an irreducible $\mathbb{K}$-invariant kernel. An explicit construction of the characteristic function of a $\frac{1}{K}$-contraction is provided. A characterization of a $\mathbb{K}$-invariant kernel with the complete Nevanlinna-Pick property is obtained via the existence of characteristic functions associated with $\frac{1}{K}$-contractions.