Construction of Spaces with an Indefinite two-Metric and Applications
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2025
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| _version_ | 1866909575323582464 |
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| author | Ferrer, Osmin Ferrer, Kandy Cure, Jaffeth |
| author_facet | Ferrer, Osmin Ferrer, Kandy Cure, Jaffeth |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_08239 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of Spaces with an Indefinite two-Metric and Applications Ferrer, Osmin Ferrer, Kandy Cure, Jaffeth Functional Analysis 42C15, 46C05, 46C20 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. |
| title | Construction of Spaces with an Indefinite two-Metric and Applications |
| topic | Functional Analysis 42C15, 46C05, 46C20 |
| url | https://arxiv.org/abs/2504.08239 |