Composition operators on Hardy space in several complex variables

Abstract

Composition operators in H* originally appeared in the work of Koopman [5] on classical mechanics, and they have played a role in ergodic theory [3]. These operators also appear in a natural way in spaces with reproducing kernel, for example, by characterization of the automorphisms of the algebra of analytic functions in B, and continuous on B,,, by studying of cornmutant of Toeplitz operators [ 11. Nordgren in his work [6] gave a brief review of the results obtained in this field. The main purpose of this exposition is to study some properties of the composition operators on Hardy space of analytic functions in the unit ball B, = {z E c”; IzI < 1 }. More precisely: first we shall characterize the composition operators’ class and the invertibility of these operators. Next we shall present examples of unbounded composition operators on H’(l3,) and prove some necessary and sufficient conditions for boundedness of composition operators. Later we shall prove three geometrical conditions necessary for the compactness of a composition operator. In the last two sections we shall give a simple characterization of the Hilbert-Schmidt composition operators, a sufficient condition for a composition operator to be trace class, and a formula for computing the trace of such an operator,