Construction of Spaces with an Indefinite two-Metric and Applications
Osmin Ferrer, Kandy Ferrer, Jaffeth Cure
TL;DR
The paper develops the theory of two-Krein spaces by constructing spaces with an indefinite two-metric from a classical Krein space, and shows that essential structural features such as orthogonality and the fundamental symmetry transfer to the standardized setting. It defines an indefinite two-metric $\psi$ and the associated $\mathcal{J}$-two-inner product $\psi_{\mathcal{J}}$, establishing norm equivalence across different decompositions and ensuring a coherent two-norm framework. It further introduces $t$-variation for functions in standardized two-Krein spaces, proves the decomposition-independence of this variation, and constructs a BV class with a two-norm, linking back to the classical bounded variation in Krein spaces as a special case. Overall, the work broadens the functional-analytic toolkit for spaces with indefinite two-inner products, providing a bridge from Krein-space theory to the analysis of indefinite two-metrics and their associated variation concepts.
Abstract
In this work, we introduce the notion of a two-Krein space and show that, starting from any classical Krein space, it is possible to construct spaces endowed with an indefinite two-inner product (admitting both positive and negative values). We develop the theory of two-Krein spaces, extending the classical structure and providing new tools for analysis in spaces with an indefinite two-inner product. It is established that the fundamental decomposition of a Krein space transfers orthogonality to the space with an indefinite two-metric. Moreover, the properties of the fundamental symmetry of the classical Krein space are carried over to the standardized two-Krein space. It is shown that the sets of positive and negative vectors generate a space with a semi-definite positive and complete two-inner product. One of the most important results in the theory of spaces with an indefinite metric is the equivalence of norms; in this work, we extend this result to standardized spaces with an indefinite two-metric. Additionally, the notion of function strongly of bounded variation in two-Krein spaces is introduced and some of their properties are established. It is also shown that the classical definition of bounded variation in two-Hilbert spaces is a particular case of the one presented in this work. Furthermore, we present a technique to construct functions of bounded t-variation in standardized two-Krein spaces from functions of bounded variation in Krein spaces, and we guarantee that when the t-variation of a function is zero, the two-norm evaluated at the images of the function remains constant with respect to t. Finally, we show that the class of strongly bounded t-variation functions in a standardized two-Krein space can be endowed with the structure of a two-norm.