More classes of non-orbit-transitive operators
Tom 197 / 2010
Streszczenie
In \cite{JKP} and its sequel \cite{FPS} the authors initiated a program whose (announced) goal is to eventually show that no operator in ${\cal L}({\cal H})$ is orbit-transitive. In \cite{JKP} it is shown, for example, that if $T\in{\cal L}({\cal H})$ 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 \cite{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 \cite{JKP}. Our main result herein is that if $T$ belongs to a certain class of $2$-normal operators in $\mathcal{L(H}^{(2)})$ and there exist two constants $\delta,\rho>0$ satisfying $\|T^{k}\|_{e}>\rho k^{\delta}$ for all $k\in\mathbb{N}$, 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.
