On numerically hypercyclic operators

Abstract

According to Kim, Peris and Song, a continuous linear operator T on a complex Banach space X is called numerically hypercyclic if the numerical orbit tf pT n xq : n P Nu is dense in C for some x P X and f P X ˚satisfying }x} " }f } " f pxq " 1. They have characterized numerically hypercyclic weighted shifts and provided an example of a numerically hypercyclic operator on C 2 .We answer two questions of Kim, Peris and Song. Namely, we construct a numerically hypercyclic operator, whose square is not numerically hypercyclic as well as an operator which is not numerically hypercyclic but has two numerical orbits whose union is dense in C. We characterize numerically hypercyclic operators on C 2 as well as the operators similar to a numerically hypercyclic one and those operators whose conjugacy class consists entirely of numerically hypercyclic operators. We describe in spectral terms the operator norm closure of the set of numerically hypercyclic operators on a reflexive Banach space. Finally, we provide criteria for numeric hypercyclicity and decide upon the numerical hypercyclicity of operators from various classes.

Cesàro bounded operators in Banach spaces

“…Also by [1, Theorem 1], there exist forward weighted shifts on ℓ 2 (N) that are strict m-isometries for m ≥ 2. Now, using that if 1 < p < ∞ and T is a forward weighted shift on ℓ p (N), then T is numerically hypercyclic if and only if T is not power bounded ( [18] & [25]), we obtain the result.…”

“…In [25,Proposition 1.5], Shkarin proved that T ∈ B(H) is weakly numerically hypercyclic if and only if there exist x, y ∈ H such that the set { T n x, y : n ∈ N} is dense in C.…”

“…By [25, Theorem 1.13], if T ∈ B(C 2 ) is a weakly numerically hypercyclic operator, then there exists λ ∈ σ(T ), with |λ| > 1 and thus T is not an m-isometry. For n = 3, it is the same by[25, Theorem 1.14].…”

“…Then, since x n = T n x n is weakly convergent but it is not norm convergent, by [25,Lemma 6.1] there is y ∈ H such that {n x n , y : n ∈ N} is dense on C. Hence T is weakly numerically hypercyclic.…”