On Positive Vectors in Indefinite Inner Product Spaces

TL;DR

This paper investigates positive vectors in indefinite inner product spaces, showing how operators acting on these vectors relate to unitary operators and how to construct a compatible definite inner product.

Contribution

It establishes a unique determination of linear operators by their action on positive vectors and links semi-groups to unitary groups via inner product transformations.

Findings

Bijectivity on positive vectors implies closeness to a unitary operator.

Semi-groups of positive vector mappings can be transformed into unitary groups.

Constructing a definite inner product ensures unitarity of the transformed operators.

Abstract

Let $\mathcal{H}$ be a linear space equipped with an indefinite inner product $[\cdot, \cdot]$. Denote by $\mathcal{F}_{++}=\{f\in\mathcal{H} \ : \ [f,f]>0\}$ the nonlinear set of positive vectors in $\mathcal{H}$. We demonstrate that the properties of a linear operator $W$ in $\mathcal{H}$ can be uniquely determined by its restriction to $\mathcal{F}_{++}$. In particular, we prove that the bijectivity of $W$ on $\mathcal{F}_{++}$ is equivalent to $W$ being {\em close} to a unitary operator with respect to $[\cdot, \cdot]$. Furthermore, we consider a one-parameter semi-group of operators $W_+ = \{W(t) : t \geq 0\}$, where each $W(t)$ maps $\mathcal{F}_{++}$ 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 $U = \{U(t) : t\in\mathbb{R}\}$ of operators that are unitary with respect to $[\cdot, \cdot]$. By imposing additional conditions, we show how to construct a suitable definite inner product $\langle\cdot, \cdot\rangle$, based on $[\cdot, \cdot]$, which guarantees the unitarity of the operators $U(t)$ in the Hilbert space obtained by completing $\mathcal{H}$ with respect to $\langle\cdot, \cdot\rangle$.