More classes of non-orbit-transitive operators

Carl Pearcy ¹   ; Lidia Smith ¹

¹ Department of Mathematics Texas A&M University College Station, TX 77843, U.S.A.

Studia Mathematica, Tome 197 (2010) no. 1, pp. 43-55

Résumé

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.

Détail

DOI : 10.4064/sm197-1-4

Keywords: cite jkp its sequel cite fps authors initiated program whose announced eventually operator cal cal orbit transitive cite jkp shown example cal cal essential calkin norm equal its essential spectral radius compact perturbation orbit transitive nbsp cite fps result extended say element class operators weakly orbit transitive present note compact perturbation certain normal operators which general satisfy orbit transitive answers question raised cite jkp main result herein belongs certain class normal operators mathcal there exist constants delta rho satisfying rho delta mathbb every compact operator operator orbit transitive seems first result yields non orbit transitive operators which modest growth rate sufficient orbit tending infinity