The order of neutrality for linear operators on inner product spaces
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.
scientific article; zbMATH DE number 1044224