Woracek, Harald Directing functionals and de Branges space completions in almost Pontryagin spaces. (English) Zbl 1395.46019 Colombo, Fabrizio (ed.) et al., Advances in complex analysis and operator theory. Festschrift in honor of Daniel Alpay’s 60th birthday. Contributions partly based on the presentations at the international conference on complex analysis and operator theory, Chapman University, Orange, CA, USA, November 2016. Cham: Birkhäuser/Springer (ISBN 978-3-319-62361-0/hbk; 978-3-319-62362-7/ebook). Trends in Mathematics, 347-398 (2017). Summary: The following theorem holds: Let \(\mathcal L\) 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 \(\mathcal L\). If \(S\) possesses a universal directing functional \(\Phi:\mathcal{L} \times \mathbb{C\;\rightarrow\mathbb{C}}\) which is real w.r.t. the given involution, and the closure of \(S\) in the completion of \(\mathcal L\) has defect index \((1, 1)\), then there exists a de Branges (Hilbert-) space \(\mathcal{B}\) such that \(x\mapsto \Phi(x,\cdot)\) maps \(\mathcal{L}\) isometrically onto a dense subspace of \(\mathcal{B}\) and the multiplication operator in \(\mathcal{B}\) is the closure of the image of \(S\) under this map.{ }In this paper, we consider a version of universal directing functionals defined on an open set \(\Omega\subseteq \mathbb{C}\) instead of the whole plane, and inner product spaces \(\mathcal{L}\) having finite negative index. We seek for representations of \(S\) in a class of reproducing kernel almost Pontryagin spaces of functions on \(\Omega\) having de Branges-type properties. Our main result is a version of the above stated theorem, which gives conditions making sure that \(\Phi\) 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 \(\Omega\) has a reproducing kernel almost Pontryagin space completion, then this completion is a de Branges-type almost Pontryagin space. This is an important fact in applications. The corresponding result in the case that \(\Omega=\mathbb{C}\) and \(\mathcal{L}\) is positive semidefinite is well known, often used, and goes back (at least) to work of M. Riesz [Ark. Mat. Astron. Fys. 17, No. 16, 52 p. (1923; JFM 49.0195.01)] .For the entire collection see [Zbl 1381.00024]. Cited in 1 Document MSC: 46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) 47B50 Linear operators on spaces with an indefinite metric 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) Keywords:directing functional; almost Pontryagin space; reproducing kernel space; de Branges space; completion Citations:JFM 49.0195.01 Software:CliffSymNil; StableBBasisNBM5; RAGlib; CliffMath; Genius; CliffOC PDFBibTeX XMLCite \textit{H. Woracek}, in: Advances in complex analysis and operator theory. Festschrift in honor of Daniel Alpay's 60th birthday. Contributions partly based on the presentations at the international conference on complex analysis and operator theory, Chapman University, Orange, CA, USA, November 2016. 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