Hilbert Space Factorization

Gustafsson, B; Putinar, M

doi:10.1007/978-3-319-65810-0_3

Hilbert Space Factorization

Lookup NU author(s): Professor Mihai Putinar

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Abstract

© Springer International Publishing AG 2017. The positivity properties of the exponential transform define in a canonical way a Hilbert space ℋ(Ω) and a (co)hyponormal operator acting on it, such that the exponential transform itself appears as a polarized compressed resolvent of this operator. There are many variants of this procedure, but they are all equivalent. Historically the process actually went in the opposite direction starting with an abstract Hilbert space and hyponormal operator with rank one self-commutator, the exponential transform arose as a natural characteristic function obtained as the determinant of the multiplicative commutator of the resolvent. If one considers ℋ(Ω) as a function space, the analytic functions in it are very weak, i.e., have a small norm, and for a special class of domains, the quadrature domains, the analytic subspace, ℋa(Ω), even collapses to a finite dimensional space. Some more general kinds of quadrature domains are discussed in terms of analytic functionals, and we also show that some integral operators based on the exponential transform can be interpreted in terms of Silva-Köthe-Grothendieck duality.