Abstract. In this paper we investigate composition operators induced on the Dirichlet space by linear fractional maps. We characterize the essential normality in this setting, obtain conditions for the linear fractional symbols j and y of the unit disc for which C_j C_y ^* or C_y ^*C_j   is compact, and investigate the shape of the numerical range for linear fractional composition operators induced on the Dirichlet space.

Keywords:   Composition operators, compact composition operators, Dirichlet space, essentially normal operators, numerical range.

AMS Subject classification:  Primary: 47B33; Secondary: 30H05.

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