We show that a continuous linear operator T on a Fre chet space satisfies the so-called Hypercyclicity Criterion if and only if it is hereditarily hypercyclic, and if and only if the direct sum TÄ T is hypercyclic. In particular, hypercyclic operators with either a dense generalized kernel or a dense set of periodic points (i.e.
We introduce a notion of disjointness for finitely many hypercyclic operators acting on a common space, notion that is weaker than Furstenberg's disjointness of fluid flows. We provide a criterion to construct disjoint hypercyclic operators, that generalizes some well-known connections between the Hypercyclicity Criterion, hereditary hypercyclicity and topological mixing to the setting of disjointness in hypercyclicity.We provide examples of disjoint hypercyclic operators for powers of weighted shifts on a Hilbert space and for differentiation operators on the space of entire functions on the complex plane.
We generalize the notions of hypercyclic operators, U-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called A-frequent hypercyclicity. We then state an A-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the A-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets N (x, U ) = {n ∈ N : T n x ∈ U } for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.
Chan and Shapiro showed that each (non-trivial) translation operator f (z) T λ → f (z + λ) acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. We show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. We also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C 0 -semigroups. We also provide an easy proof of the result of Salas that every infinitedimensional Banach space supports arbitrarily large tuples of dual d-hypercyclic operators, and construct an example of a mixing Hilbert space operator T so that (T , T 2 ) is not d-mixing.
We characterize disjoint hypercyclicity and disjoint supercyclicity of finitely many linear fractional composition operators acting on spaces of holomorphic functions on the unit disc, answering a question of Bernal-González. We also study mixing and disjoint mixing behavior of projective limits of endomorphisms of a projective spectrum. In particular, we show that a linear fractional composition operator is mixing on the projective limit of the S v spaces strictly containing the Dirichlet space if and only if the operator is mixing on the Hardy space.
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