Essential normality of linear fractional composition operators in the unit ball of \(\mathbb C^N\) (Q983672)

An operator \(T\) defined on a Hilbert space is called essentially normal if \([T,T^*]=TT^*-T^*T\) is compact. The authors study the essential normality of certain linear fractional composition operators \(C_\phi\) acting on Hardy and weighted Bergman spaces defined in the unit ball \(B_n\) of \(\mathbb C^n\). Their results extend those by \textit{B.\,D.\thinspace MacCluer} and \textit{R.\,J.\thinspace Weir} [``Linear-fractional composition operators in several variables'', Integral Equations Oper.\ Theory 53, No.\,3, 373--402 (2005; Zbl 1121.47017)]. A basic formula on \([C^*_\phi, C_\phi]\), leads to the study of the compactness of \([T^*_{z^\beta},C_\phi]\) where \(\phi\) is a linear fractional map of \(B_n\). One basic estimate shown in the paper which is used in the proof of the main result is that \(1- \Re \phi_1(z)\approx \Re (1-z_1)\) holds true in a neighborhood of \(e_1\) on \(\bar B_n\) for any linear fractional self-map with \(\phi(e_1)=e_1\), where \(\phi_1\) and \(z_1\) are the first coordinate of \(\phi\) and \(z\). In the particular cases of parabolic or hyperbolic fractional maps of \(B_2\), they are able to characterize the essential normality of composition operators.