On the linear dynamics of shift operators on (PLB)-spaces

Abstract In this paper we investigate dynamical properties—such as (sequential) hypercyclicity, topological transitivity, topological mixing, and topological ergodicity—of backward shift and weighted pseudoshift operators acting on countable projective limits of countable inductive limits of Banach spaces associated with a Schauder basis. In the second part, we analyze the linear dynamics of backward shift and weighted pseudoshift operators on projective limits of Köthe co-echelon spaces determined by a system of weights $$a_p(\mathcal {V})$$ a p ( V ) , $$1 \le p < \infty $$ 1 ≤ p < ∞ . We also provide an explicit example of a weight system $$\mathcal {V}$$ V ( $$\mathcal {V}'$$ V ′ ) for which the backward shift is mixing and chaotic on the sequence (PLB)-spaces $$a_p(\mathcal {V})$$ a p ( V ) (on the sequence (LF)-spaces $$l_p(\mathcal {V}')$$ l p ( V ′ ) ), while it fails to be well defined on $$l_p(\mathcal {V})$$ l p ( V ) (on $$a_p(\mathcal {V}')$$ a p ( V ′ ) ). Finally, we examine backward shift operators on weighted versions of the classical sequence spaces $${{\,\textrm{ces}\,}}(p)$$ ces ( p ) , d ( p ) and $$\ell ^p$$ ℓ p , and on their corresponding versions of (PLB)- and (LF)-spaces.