More classes of non-orbit-transitive operators

Carl Pearcy ; Lidia Smith

Studia Mathematica, Tome 196 (2010), p. 43-55 / Harvested from The Polish Digital Mathematics Library

Résumé

In [JKP] and its sequel [FPS] the authors initiated a program whose (announced) goal is to eventually show that no operator in ℒ(ℋ) is orbit-transitive. In [JKP] it is shown, for example, that if T ∈ ℒ(ℋ) and the essential (Calkin) norm of T is equal to its essential spectral radius, then no compact perturbation of T is orbit-transitive, and in [FPS] this result is extended to say that no element of this same class of operators is weakly orbit-transitive. In the present note we show that no compact perturbation of certain 2-normal operators (which in general satisfy $||T||_{e} > r_{e}(T)$) can be orbit-transitive. This answers a question raised in [JKP]. Our main result herein is that if T belongs to a certain class of 2-normal operators in $ℒ(ℋ^{(2)})$ and there exist two constants δ,ρ > 0 satisfying $||T^{k}||_{e} > ρk^{δ}$ for all k ∈ ℕ, then for every compact operator K, the operator T+K is not orbit-transitive. This seems to be the first result that yields non-orbit-transitive operators in which such a modest growth rate on $||T^{k}||_{e}$ is sufficient to give an orbit ${T^{k}x}$ tending to infinity.

Détail

Publié le : 2010-01-01

Zbl 1190.47009

EUDML-ID : urn:eudml:doc:286347