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Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\calH}$ 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 $$ {\calI}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ Itis proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra ofadjointable operators, and a homogeneous space of the group of invertibleoperators in ${\cal H}$, which are unitaries for the $A$-inner product. Smoothcurves in ${\cal I}_A^a$ with given initial conditions, which are minimal forthe metric induced by $\langle \ , \ \rangle_A$, are presented. This resultdepends on an adaptation of M.G. Krein's extension method of symmetriccontractions, in order that it works also for symmetrizable transformations(i.e., operators which are selfadjoint for the $A$-inner product).

Operators which preserve a positive definite inner product