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More classes of non-orbit-transitive operators

100%

Pearcy C., Smith L.

Studia Mathematica

2010

tom 197

nr 1

43-55

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.

Quasiaffine transforms of operators

81%

Jung I., Ko E., Pearcy C.

Studia Mathematica

2009

tom 193

nr 3

263-267

We obtain a new sufficient condition (which may be useful elsewhere) that a compact perturbation of a normal operator be the quasiaffine transform of some normal operator. We also give some applications of this result.

Ograniczanie wyników

2 Studia Mathematica

2 Pearcy C.

1 Jung I.

1 Ko E.

1 Smith L.

1 2010

1 2009

Wyniki wyszukiwania

More classes of non-orbit-transitive operators

Quasiaffine transforms of operators