Erratum to: PF property and property $$(\beta )$$ ( β ) for paranormal operators

Abstract

The proof of Theorem 7.2(ii)—“If A ∈ B(H) ∩ PF(δ) is a contraction, then S(δA,V ∗) = ∅ implies A is not n-supercyclic”—of our paper of the title is incomplete (in that it fails to consider the case α(A∗c − λ) = 0). We provide here additional argument to complete the proof, and prove an analogue of the result for weakly supercyclic operators. We follow the notation and terminology of the paper of the title. Thus B(H) denotes the algebra of bounded linear operators on an infinite dimensional complexHilbert space, δA,B ∈ B(B(H)) denotes the generalized derivation δA,B(X) = AX − X B, and an operator A ∈ B(H) satisfies thePutnam–Fuglede property δ, denoted A ∈ PF(δ), if whenever the equation AX = X V ∗ holds for some isometry V and operator X ∈ B(H), then also A∗X = X V . An operator A ∈ B(H) is n-supercyclic for some n ∈ N if H has an n-dimensional subspace M with dense orbit OrbM (A) = ⋃m∈N Am M ; a 1-supercyclic operator is supercyclic, and we say that A is weakly supercyclic if there exists a vector x ∈ H, with M the corresponding one dimensional subspace generated by x , such that OrbM (A) is weakly dense (i.e., dense in the weak topology) inH. It is clear that if an operator A ∈ B(H) is n-supercyclic or weakly supercyclic, then H is separable (see [1,2,4,6] for more information). For an operator A ∈ B(H), let S(δA,V ∗) = {V ∈ B(H) : V isometric, δ−1 A,V ∗(0) = {0}}, and let SD(δA,V ∗) denote the set of those isometries V ∈ B(H) for which there exists an X ∈ B(H) with dense range such that δA,V ∗(X) = 0. Clearly, if A ∈ B(H) is a contraction,