Dilations of Γ-contractions by solving operator equations

Bhattacharyya, Tirthankar ; Pal, Sourav ; Shyam Roy, Subrata (2012) Dilations of Γ-contractions by solving operator equations Advances in Mathematics, 230 (2). pp. 577-606. ISSN 0001-8708

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.aim.2012.02.016


For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equation S-S*P = (I-P*P)1/2 X(I-P*P) 1/2, where X is a bounded operator on Ran(I-P*P) 1/2 with numerical radius of X being not greater than 1. A pair of bounded operators (S,P) which has the domain Γ = {(z1+z2, z1z2):|z1| ≤ 1,|z2| ≤ 1} ⊆ C2 as a spectral set, is called a Γ-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Γ-contraction (S,P). This allows us to construct an explicit Γ-isometric dilation of a Γ-contraction (S,P). We prove the other way too, i.e, for a commuting pair (S,P) with |P|| ≤ 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S,P) is a Γ-contraction. We show that for a pure Γ-contraction (S,P), there is a bounded operator C with numerical radius not greater than 1, such that S = C +C*P. Any Γ-isometry can be written in this form where P now is an isometry commuting with C and C*. Any Γ-unitary is of this form as well with P and C being commuting unitaries. Examples of Γ-contractions on reproducing kernel Hilbert spaces and their Γ-isometric dilations are discussed.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Gamma Contractions; Spectral Set; Gamma Isometric Dilation; Fundamental Equation
ID Code:99741
Deposited On:27 Nov 2016 12:50
Last Modified:29 Nov 2016 06:25

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