Operators which preserve a positive definite inner product

2021/10/19 by Esteban Andruchow, Andruchow, Esteban
Mathematics · #47A05 #47A62 #58B10 #58B20 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2110.10304

openalex publication_date 2021/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \cal H be a Hilbert space, A a positive definite operator in \cal H and ⟨ f,g⟩_A=⟨ Af,g⟩, f,g∈ \cal H, the A-inner product. This paper studies the geometry of the set \cal I_A^a:=\\hbox adjointable isometries for ⟨ , ⟩_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 ⟨ , ⟩_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).

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