Standard symmetric operators in Pontryagin spaces: A generalized von Neumann formula and minimality of boundary coefficients
type: Item
instance of: scholarly article
title: Standard symmetric operators in Pontryagin spaces: A generalized von Neumann formula and minimality of boundary coefficients
zbMATH Open document ID: 1028.47027
DOI: 10.1016/S0022-1236(02)00041-1
author: Branko Ćurgus
author: Tomas Ya. Azizov
author: A. Dijksma
published in: Journal of Functional Analysis
publication date: 2003-04-28T00:00:00Z
review text: A symmetric operator \(S\) in a Pontryagin space is used to characterize the boundary coefficients, which are meromorphic matrix functions on \(\mathbb{C}\setminus\mathbb{R}\). If \(Q\) is an invertible self-adjoint matrix, then a \(Q\)-boundary coefficient is a matrix valued function \({\mathcal U}\) which satisfies several conditions including the equality \({\mathcal U}(z)Q^{-1} {\mathcal U}(z^*)^*=0\), and the existence of the limit \[ \lim_{w\to z^*}{{\mathcal U}(z)Q^{-1}{\mathcal U}(w)^* \over z-w^*} \] at each \(z\in\text{dom}({\mathcal U})\). A closed symmetric relation in a \(\kappa\)-Pontryagin space is considered to be standard if for some \(\mu\in \mathbb{C}\setminus \mathbb{R}\), \(\mu\notin\sigma_p (S)\), each of the sets \(\mathbb{C}^\pm\cap \sigma_p(S)\) contains at most \(\kappa\) points. A generalization of the von Neumann's formula \[ S^{\langle *\rangle}=S +S^{\langle *\rangle} \cap\mu I+S^{ \langle *\rangle} \cap\mu^*I,\;\mu\in\mathbb{C} \setminus\mathbb{R}, \] is introduced to study a minimality property of the \(Q\)-boundary coefficients, which is expressed in terms of maximal rank of a matrix formed with the values of \({\mathcal U}\) at \(\kappa\) distinct points in \(\mathbb{C}^+\cap \text{dom} ({\mathcal U})\). The proofs require an extensive knowledge of the theory of operators in indefinite inner product spaces. Detailed explanations, partial results, and references to previous techniques are very well organized in six sections of the paper and an appendix (on Schur functions).
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 34B07
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 47A06
zbMATH DE Number: 1901774
zbMATH Keywords: boundary operators
zbMATH Keywords: self-adjoint operations and relations
zbMATH Keywords: von Neumann's equality
reviewed by: Trandafir Balan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901774