Weighted Composition Operators and Supercyclicity Criterion

The vector x is called supercyclic for T ifC orb T, x is dense inH. Also a supercyclic operator is one that has a supercyclic vector. For some sources on these topics, see 1–16 . Let H be a separable Hilbert space of functions analytic on a plane domain G such that, for each λ in G, the linear functional of evaluation at λ given by f → f λ is a bounded linear functional on H. By the Riesz representation theorem, there is a vector Kλ in H such that f λ 〈f,Kλ〉. We call Kλ the reproducing kernel at λ. A complex-valued function φ on G is called a multiplier ofH if φH ⊂ H. The operator of multiplication by φ is denoted byMφ and is given by f → φf . If φ is a multiplier of H and ψ is a mapping from G into G, then Cφ,ψ : H → H by