Operators Which Preserve a Positive Definite Inner Product

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