Year: 2009
Type: preprint
Abstract: The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel $k_S(z,w) = (1 - z\ow)^{-1}$ for $|z|, |w| < 1$, by means of $(1/k_S)(T,T^*) \ge 0$, we consider an arbitrary open connected domain $Ω$ in $\BC^n$, a complete Nevanilinna-Pick kernel $k$ on $Ω$ and a tuple $T = (T_1, ..., T_n)$ of commuting bounded operators on a complex separable Hilbert space $\clh$ such that $(1/k)(T,T^*) \ge 0$. For a complete Pick kernel the $1/k$ functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with $T$. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples $T$.
Source: arXiv (Cornell University)
Institution: Indian Institute of Science Bangalore
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2952304687