On positive vectors in indefinite inner product spaces

Let [Formula: see text] be a linear space equipped with an indefinite inner product [Formula: see text]. Denote by [Formula: see text] the nonlinear set of positive vectors in [Formula: see text]. We demonstrate that the properties of a linear operator W in [Formula: see text] can be uniquely determined by its restriction to [Formula: see text]. In particular, we prove that the bijectivity of W on [Formula: see text] is equivalent to W being close to a unitary operator with respect to [Formula: see text]. Furthermore, we consider a one-parameter semi-group of operators [Formula: see text], where each [Formula: see text] maps [Formula: see text] onto itself in a one-to-one manner. We show that, under this natural restriction, the semi-group [Formula: see text] can be transformed into a one-parameter group [Formula: see text] of operators that are unitary with respect to [Formula: see text]. By imposing additional conditions, we show how to construct a suitable definite inner product [Formula: see text], based on [Formula: see text], which guarantees the unitarity of the operators [Formula: see text] in the Hilbert space obtained by completing [Formula: see text] with respect to [Formula: see text].