Abstract Hankel Operators in Kreĭn Spaces

Abstract

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered in the Kreȋn space setting. Under a generic assumption, without which the Krein space case may be untreatable, a necessary and sufficient condition for the existence of Hankel symbols for a given Hankel operator X is given. A parametric labeling of the Hankel symbols of X by means of Schur class functions is obtained. The proof is established by associating to the data of the problem an isometry V acting on a Kreȋn space so that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of V . The result includes uniqueness criteria and a Schur like formula. Mathematics Subject Classification (2010). Primary: 47B35; Secondary: 47B50, 47A20.