Contributions to Books:

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.

English abstract:
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 1920s.

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

Electronic version of the publication:

Created from the Publication Database of the Vienna University of Technology.