Operators which preserve a positive definite inner product

Abstract

Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the $A$-inner product. This paper studies the geometry of the set $$ {\cal I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth curves in ${\cal I}_A^a$ 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 extension method of symmetric contractions, in order that it works also for symmetrizable transformations (i.e., operators which are selfadjoint for the $A$-inner product).