Complete Nevanlinna–Pick kernels and the curvature invariant

We consider a unitarily invariant complete Nevanlinna–Pick kernel denoted by s and a commuting d-tuple of bounded operators \(\varvec{T}= (T_{1}, \dots , T_{d})\) satisfying a natural contractivity condition with respect to s. We associate with \(\varvec{T}\) its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of \(\varvec{T}\). The instrument that makes this possible is the characteristic function developed in Adv Math 426:109089, 2023, https://doi.org/10.1016/j.aim.2023.109089. We present an asymptotic formula for the curvature invariant. In the special case when \(\varvec{T}\) 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 \(\varvec{T}\) specifically when its characteristic function is a polynomial.