Operators Which Preserve a Positive Definite Inner Product

Abstract

Let \(\mathcal {H}\) be a Hilbert space, A a positive definite operator in \(\mathcal {H}\) and \(\langle f,g\rangle _A=\langle Af,g\rangle \), \(f,g\in \mathcal {H}\), the A-inner product. This paper studies the geometry of the set $$\begin{aligned} \mathcal {I}_A^a:=\{\text { adjointable isometries for } \langle \ , \ \rangle _A\}. \end{aligned}$$It is proved that \(\mathcal {I}_A^a\) is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in \(\mathcal {H}\) which are unitaries for the A-inner product. Smooth curves in \(\mathcal {I}_A^a\) with given initial conditions, which are minimal for the metric induced by \(\langle \ , \ \rangle _A\), are presented. This result depends on an adaptation of M.G. Krein’s method for the lifting of symmetric contractions, in order that it works also for symmetrizable transformations (i.e., operators which are selfadjoint for the A-inner product).