Resolvents and Nevanlinna representations in several variables
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Tue, 24/04/2012 17:00 |
Nicholas Young (Leeds) |
Functional Analysis Seminar  |
L3 |
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
 with non-negative imaginary part). I will describe a higher-dimensional
analogue of Nevanlinna's theorem. The  -variable Pick class is defined to
be the set of analytic functions on the polyhalfplane  with non-negative
imaginary part; we obtain four different representation formulae for functions
in the -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  . |
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