On Positive Vectors in Indefinite Inner Product Spaces

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 $W_{+}=\{W(t):t\ge 0\}$, 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 $⟨\cdot ,\cdot ⟩$, 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 $⟨\cdot ,\cdot ⟩$.