In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias.

We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. Surprisingly, there is a converse, which roughly says that if a kernel $k$ admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain. So, what can be said if $(T_{1}, \dots,T_{d})$ is $1/k$-contractive when $k$ is an irreducible unitarily invariant kernel, but does not have the complete Nevanlinna-Pick property? We shall see that if $k$ has a complete Nevanlinna-Pick factor $s$, then much can be retrieved.

We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction specifically when its characteristic function is a polynomial.