[Back]

H. Woracek:
"Directing functionals and de Branges space completions in almost Pontryagin spaces";
in: "ASC Report 19/2016", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, 1 - 51.

The following theorem holds: Let L be a - not necessarily nondegenerated or complete - positive semidefinite inner product space carrying an anti-isometric involution, and let S be a symmetric operator in L. If S possesses a universal directing functional Φ : L × C → C which is real w.r.t. the given involution, and the closure of S in the completion of L has defect index (1, 1), then there exists a de Branges (Hilbert-) space B such that x 􀀀→ Φ(x, ·) maps L isometrically onto a dense subspace of B and the multiplication operator in B is the closure of the image of S under this map.
open set Ω ⊆ C instead of the whole plane, and inner product spaces L having finite negative index. We seek for representations of S in a class of reproducing kernel almost Pontryagin spaces of functions on Ω having de Branges-type properties. Our main result is a version of the above stated theorem, which gives conditions making sure that Φ establishes such a representation. This result is accompanied by a converse statement and some supplements.
As a corollary, we obtain that if a de Branges-type inner product space of analytic functions on Ω has a reproducing kernel almost Pontryagin space completion, then this completion is a de Branges-type almost Pontryagin space. This is an important
act in applications. The corresponding result in the case that Ω = C and L is positive semidefinite is well-known, often used, and goes back (at least) to work of M.Riesz in the 1920´s.

directing functional, almost Pontryagin space, reproducing kernel space, de Branges space, completion

http://www.asc.tuwien.ac.at/preprint/2016/asc19x2016.pdf