Dilations of \Gamma-contractions by solving operator equations

TL;DR

This paper investigates operator equations related to aGamma-contractions, establishing existence, uniqueness, and explicit dilations, thereby advancing the understanding of operator models within this spectral set framework.

Contribution

It provides new results on the existence and uniqueness of solutions to operator equations for aGamma-contractions and constructs explicit aGamma-isometric dilations, enriching the operator theory of spectral sets.

Findings

Existence and uniqueness of solutions to the operator equation for aGamma-contractions.

Explicit construction of aGamma-isometric dilations for aGamma-contractions.

Characterization of aGamma-isometries and aGamma-unitaries in terms of operators C and P.

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.