24 April 2012
A theorem of R. Nevanlinna from 1922 characterizes the Cauchy transforms of finite positive measures on the real line as the functions in the Pick class that satisfy a certain growth condition on the real axis; this result is important in the spectral theory of self-adjoint operators. (The Pick class is the set of analytic functions in the upper half-plane $\Pi$ with non-negative imaginary part). I will describe a higher-dimensional analogue of Nevanlinna's theorem. The $n$-variable Pick class is defined to be the set of analytic functions on the polyhalfplane $\Pi^n$ with non-negative imaginary part; we obtain four different representation formulae for functions in the $n$-variable Pick class in terms of the ``structured resolvent" of a densely defined self-adjoint operator. Structured resolvents are analytic operator-valued functions on the polyhalfplane with properties analogous to those of the familiar resolvent of a self-adjoint operator. The types of representation that a function admits are determined by the growth of the function on the imaginary polyaxis $(i\R)^n$.
- Functional Analysis Seminar