Abstract. In the present paper we investigate conditions under which a holomorphic self-map of the open unit disk induces a hypercyclic weighted composition operator in the space of holomorphic functions.

Suppose thatXis a separable normed space and the operatorsAandQare bounded onX. In this paper, it is shown that ifAQ=QA,Ais an isometry, andQis a nilpotent then the operatorA+Qis neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space andAis a co-isometric operator, then we give sufficient conditions under which the operatorA+Qsatisfies the supercyclicity criterion.

Abstract.Let Í2 be a bounded plane domain. Sufficient conditions are given so that an operator T in the Cowen-Douglas class ^" (ii) is reflexive. The operator Mz of multiplication by z on a Hubert space of functions analytic on a finitely connected domain Q is shown to be reflexive whenever a{Mz) = £2 is a spectral set.

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