Dilations of Γ-contractions by solving operator equations

Abstract

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.