Rearrangement Invariant Orthogonal Sums in Krein Spaces. II
Michael A. Dritschel, Alejandra Maestripieri, James Rovnyak
TL;DR
This work extends the rearrangement-invariant orthogonal sums theory from regular subspaces to the broader classes of pseudo-regular and quasi-pseudo-regular subspaces in Krein spaces. It develops Moore-Smith convergence and operator-range techniques to define and analyze infinite orthogonal sums, addressing existence, directness, and regularity issues. The results generalize Part I by providing conditions under which sums remain pseudo-regular or become qpr, and by characterizing the sums in terms of operator ranges and nets of projections. Collectively, the paper clarifies when orthogonal sums of qpr or pseudo-regular subspaces are well-defined, unique, and maintain the desired indefinite inner product structure, thereby extending Hilbert-space intuition to Krein spaces.
Abstract
Part I of the paper considered infinite orthogonal sums of regular subspaces in a Krein space (that is, of subspaces which are themselves Krein spaces). How precisely these sums should be defined and conditions for when such a sum is itself regular were examined. These included, for example, a boundedness condition for the sum of the corresponding orthogonal projections. The same problem is addressed here for (quasi-)pseudo-regular subspaces. Such subspaces happen to be the orthogonal direct sum of a regular space and an isotropic, or neutral, subspace. Alternate characterizations of such subspaces are given, along with various examples and counter-examples. The infinite orthogonal sum of these is then considered.