A bounded linear operator on a separable Banach space is said to satisfy the three-neighborhood condition if for every pair U, V of non-empty open subsets of X, and each open neighborhood W of zero in X there exists a positive integer n such that both TnU ∩ W and TnW ∩ V are non-empty. The operator is called syndetically hypercyclic if for any strictly increasing syndetic sequence of positive integers {n k}k, { Tnk }k is a hypercyclic sequence of operators. We prove that these two conditions are equivalent to the Hypercyclicity Criterion. Then we prove the existence of topologically transitive multipliers on Banach algebras and study some necessary conditions for a multiplier to be topologically transitive on Banach algebras.
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