Dilations of \Gamma-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 \Gamme = {(z 1 +z 2, z 1z 2) : |z1|{\leq} 1, |z2| {\leq}1} {\subseteq} C2 as a spectral set, is called a \Gamme-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a \Gamma-contraction (S,P). This allows us to construct an explicit \Gamma-isometric dilation of a \Gamma-contraction (S,P). We prove the other way too, i.e, for a commuting pair (S,P) with |P|| {\leq} 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 \Gamma-contraction. We show that for a pure \Gamma-contraction (S,P), there is a bounded operator C with numerical radius not greater than 1, such that S = C +C*P. Any \Gamma-isometry can be written in this form where P now is an isometry commuting with C and C*. Any \Gamma-unitary is of this form as well with P and C being commuting unitaries. Examples of \Gamma-contractions on reproducing kernel Hilbert spaces and their \Gamma-isometric dilations are discussed.