Operators which preserve a positive definite inner product

TL;DR

This paper explores the geometric structure of operators that preserve a positive definite inner product on a Hilbert space, revealing their manifold and homogeneous space properties, and characterizing minimal smooth curves within this set.

Contribution

It establishes that the set of $A$-adjointable isometries forms a submanifold and a homogeneous space, and adapts Krein's extension method for symmetrizable transformations.

Findings

${ m I}_A^a$ is a submanifold of the Banach algebra of adjointable operators.

${ m I}_A^a$ is a homogeneous space of the group of $A$-unitary invertible operators.

Minimal smooth curves in ${ m I}_A^a$ are characterized for the metric induced by the $A$-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).