Linear fractional composition operators on the Dirichlet space in the unit ball

We investigate the adjoints of linear fractional composition operators C _ϕ acting on classical Dirichlet space D(B _N ) in the unit ball B _N of ℂ ^N , and characterize the normality and essential normality of C _ϕ on D(B _N ) and the Dirichlet space modulo constant function D ₀(B _N ), where ϕ is a linear fractional map ϕ of B _N . In addition, we also show that for any non-elliptic linear fractional map ϕ of B _N , the composition maps σ o ϕ and ϕ o σ are elliptic or parabolic linear fractional maps of B _N . This work was supported by National Natural Science Foundation of China (Grant Nos. 10671141, 10371091)