ACTA issues

Pontryagin spaces of entire functions. VI

Michael Kaltenbäck, Harald Woracek

Acta Sci. Math. (Szeged) 76:3-4(2010), 511-560

Abstract. In the theory of two-dimensional canonical (also called `Hamiltonian') systems, the notion of the Titchmarsh--Weyl coefficient associated to a Hamiltonian function plays a vital role. A cornerstone in the spectral theory of canonical systems is the Inverse Spectral Theorem due to Louis de Branges which states that the Hamiltonian function of a given system is (up to changes of scale) fully determined by its Titchmarsh--Weyl coefficient. Much (but not all) of this theory can be viewed and explained using the theory of entire operators due to Mark G. Kre?n. Motivated from the study of canonical systems or Sturm--Liouville equations with a singular potential, and from other developments in the indefinite world, it was a long-standing open problem to find an indefinite (Pontryagin space) analogue of the notion of canonical systems, and to prove a corresponding analogue of de Branges' Inverse Spectral Theorem. We gave a definition of an indefinite analogue of a Hamiltonian function and elaborated the operator theory of such `indefinite canoncial systems' in previous work. In the present paper we prove the corresponding version of the Inverse Spectral Theorem.

AMS Subject Classification (1991): 34A55, 46C20, 46E22, 30H05

Keyword(s): canonical system, Pontryagin space, Inverse Spectral Theorem

Received August 4, 2009. (Registered under 91/2009.)