The order of neutrality for linear operators on inner product spaces
type: Item
instance of: scholarly article
title: The order of neutrality for linear operators on inner product spaces
zbMATH Open document ID: 0939.47032
DOI: 10.1016/S0024-3795(96)00239-X
author: P. Lancaster
author: A. Markus
author: Peter Zizler
published in: Linear Algebra and its Applications
publication date: 2000-07-05T00:00:00Z
review text: On a complex vector space \({\mathcal H}\) an inner product \([\cdot,\cdot]\) and a symmetric linear operator \(A\) are defined in a usual way. A subspace \({\mathcal S}\subseteq{\mathcal H}\) is said to be neutral if \([x,y]= 0\) for all \(x,y\in{\mathcal S}\). Earlier, Lancaster et al. proved that when \(H\) is finite-dimensional, all maximal \(A\)-invariant, neutral subspaces have the same dimension, see \textit{P. Lancaster}, \textit{A. S. Markus} and \textit{Q. Ye} [Linear Algebra Appl. 197-198, 3-29 (1994; Zbl 0793.15002)]. This result of Lancaster et al. relies on spectral properties of \(A\) and does not lend itself to the context of infinite-dimensional spaces. A new proof is devised in the reviewed contribution and depends only on the geometry of subspaces. This admits a significant generalization of the theory. The theorem proved is applied to Pontryagin spaces a particular class of Krein spaces. The note is valuable and reads well.
reviewed by: J. J. Telega
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 1044224
zbMATH Keywords: Pontryagin spaces
zbMATH Keywords: Krein spaces
MaRDI profile type: Publication
cites work: Q4403356
cites work: Low rank perturbations of strongly definitizable transformations and matrix polynomials
cites work: Invariant Neutral Subspaces for Symmetric and Skew Real Matrix Pairs
cites work: Minimal symmetric factorizations of symmetric real and complex rational matrix functions
cites work: Norms of sampling operators
cites work: Definitizable Operators on a Krein Space
Wikidata QID: Q127098026
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description: scientific article; zbMATH DE number 1044224