CHAOTIC PROPERTY OF WEIGHTED COMPOSITION OPERATORS

Abstract

Abstract. In the present paper, we study the chaotic property of wei-ghted composition operators acting on the holomorphic function space H (U). 1. IntroductionA continuous linear operator T acting on a separable Frechet space X iscalled hypercyclic provided there exists a vector x 2 X such that its orbitorb( T;x ) = fT n x : n = 0 ; 1 ;:::g is dense in X . A periodic point for T is avector x2 X such that T n x = x for some n2 N. Finally, T is said to bechaotic if it is hypercyclic and its set of periodic points is dense in X .In 1929, ff [1] showed that the translation operator T a : H (C) ! H (C)de ned by ( T a f )( z ) = f ( z + a ), a= 0, is hypercyclic on the Frechet space H (C)of entire functions. This result was generalized by Godefroy and Shapiro [7]who proved that each operator on H (C N ) which commutes with all translationsand is not a scalar multiple of the identity, is chaotic. Other classical examplesof hypercyclic and chaotic operators are weighted shifts on l p spaces [8, 10] andadjoints of multiplication operators on Hilbert spaces of holomorphic functions[7].Let U stand for the open unit disk in C. Each