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 $\langle\cdot, \cdot\rangle$.
Categories: math.FA math-ph math.MP