H. Woracek:

"Reproducing kernel almost Pontryagin spaces";

in: "ASC Report 14/2014", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2014, ISBN: 978-3-902627-07-0, 1 - 40.

An almost Pontryagin space A is an inner product space which admits a

direct and orthogonal decomposition of the form A = A>[+_ ]A with a Hilbert space A> and a nite-dimensional negative semide nite space A . A reproducing kernel almost Pontryagin space is an almost Pontryagin space of functions (de ned on some nonempty set and taking values in

some Krein space), with the property that all point evaluation functionals are continuous. We adress two problems.

1 In the presence of degeneracy, it is not possible to reproduce function values as inner products with a kernel function in the usual way. We obtain a na tural substitute for a kernel function, and study the relation between spaces and kernels in detail.

2 Given an inner product space L of functions, does there exist a reproducing kernel almost Pontryagin space A which contains L isometrically? We characterise those spaces where the answer is \yes". We show that, in case of existence, there is a

unique such space A which contains L isometrically and densely. Its geometry, in particular its degree of degeneracy, is an important invariant of L. It plays a role in connection with Krein's formula on the description of generalised resolvents and,

thus, in several concrete problems related with the extension theory of symmetric operators.

inde nite inner product, almost Pontryagin space, reproducing kernel, completion, degenerate inner product

http://www.asc.tuwien.ac.at/preprint/2014/asc14x2014.pdf

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