More classes of non-orbit-transitive operators

Carl Pearcy, and Lidia Smith. "More classes of non-orbit-transitive operators." Studia Mathematica 197.1 (2010): 43-55. <http://eudml.org/doc/286347>.

@article{CarlPearcy2010,
abstract = {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.},
author = {Carl Pearcy, Lidia Smith},
journal = {Studia Mathematica},
language = {eng},
number = {1},
pages = {43-55},
title = {More classes of non-orbit-transitive operators},
url = {http://eudml.org/doc/286347},
volume = {197},
year = {2010},
}

TY - JOUR
AU - Carl Pearcy
AU - Lidia Smith
TI - More classes of non-orbit-transitive operators
JO - Studia Mathematica
PY - 2010
VL - 197
IS - 1
SP - 43
EP - 55
AB - 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.
LA - eng
UR - http://eudml.org/doc/286347
ER -