On Positive Vectors in Indefinite Inner Product Spaces

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 $\langle\cdot, \cdot\rangle$.