Contributions to Books:
"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,
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
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.