2023/02/27 by B. V. Rajarama Bhat, Bhat, B. V. Rajarama, Anindya Ghatak +3
Mathematics · #42A70 #44A60 #47A12 #47A20 #47A57 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2302.13873
openalex publication_date 2023/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a complex Hilbert space and let \An\n≥ 1 be a sequence of bounded linear operators on H. Then a bounded operator B on a Hilbert space K ⊇ H is said to be a dilation of this sequence if An = PHBn|H for all n≥ 1, where PH is the projection of K onto H. The question of existence of dilation is a generalization of the classical moment problem. We recall necessary and sufficient conditions for the existence of self-adjoint, isometric and unitary dilations and present block operator representations for these dilations. For instance, for self-adjoint dilations one gets block tridiagonal representations similar to the classical moment problem. Given a positive invertible operator A, an operator T is said to be in the CA-class if the sequence \A-(1)/(2)TnA-(1)/(2):n≥ 1\ admits a unitary dilation. We identify a tractable collection of CA-class operators for which isometric and unitary dilations can be written down explicitly in block operator form. This includes the well-known ρ-dilations for positive scalars. Here the special cases ρ=1 and ρ=2 correspond to Schäffer representation for contractions and Ando representation for operators with numerical radius not more than one respectively.