2024/11/06 by Fabio Bagarello, Bagarello, Fabio, Sergiusz Kużel +1
Decision Sciences · Mathematics · #46C20 #47B50 #47D03 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Fuzzy and Soft Set Theory #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2411.04247
openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a linear space equipped with an indefinite inner product [⋅, ⋅]. Denote by F++=\f\inH : [f,f]>0\ the nonlinear set of positive vectors in H. We demonstrate that the properties of a linear operator W in H can be uniquely determined by its restriction to F++. In particular, we prove that the bijectivity of W on F++ is equivalent to W being \em close to a unitary operator with respect to [⋅, ⋅]. Furthermore, we consider a one-parameter semi-group of operators W+ = \W(t) : t ≥ 0\, where each W(t) maps F++ onto itself in a one-to-one manner. We show that, under this natural restriction, the semi-group W+ can be transformed into a one-parameter group U = \U(t) : t∈ℝ\ of operators that are unitary with respect to [⋅, ⋅]. By imposing additional conditions, we show how to construct a suitable definite inner product ⟨⋅, ⋅⟩, based on [⋅, ⋅], which guarantees the unitarity of the operators U(t) in the Hilbert space obtained by completing H with respect to ⟨⋅, ⋅⟩.