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  1. 3 Studia Mathematica

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  1. 3 Cao P.

  2. 1 Sun S.

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Quasinilpotent operators in operator Lie algebras II

100%
Cao P.
Studia Mathematica
|
2009
|
tom 195
|
nr 2
193-200
EN
In this paper, it is proved that the Banach algebra $\overline{𝓐(ℒ)}$, generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\overline{𝓐(ℒ)}$ consists of polynomially compact operators. It is also proved that $\overline{𝓐(ℒ)}$ consists of quasinilpotent operators if ℒ is an essentially nilpotent Engel Lie algebra generated by quasinilpotent operators. Finally, Banach algebras generated by essentially nilpotent Lie algebras are shown to be compactly quasinilpotent.
2
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Scattered elements of Banach algebras

100%
Cao P.
Studia Mathematica
|
2013
|
tom 214
|
nr 2
195-200
EN
A scattered element of a Banach algebra 𝓐 is an element with at most countable spectrum. The set of all scattered elements is denoted by 𝓢(𝓐). The scattered radical $𝓡_{sc}(𝓐)$ is the largest ideal consisting of scattered elements. We characterize in several ways central elements of 𝓐 modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) 𝓢(𝓐) + 𝓢(𝓐) ⊂ 𝓢(𝓐); (ii) 𝓢(𝓐)𝓢(𝓐) ⊂ 𝓢(𝓐); (iii) $[𝓢(𝓐),𝓐] ⊂ 𝓡_{sc}(𝓐)$.
3
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Lie algebras generated by Jordan operators

63%
Cao P., Sun S.
Studia Mathematica
|
2008
|
tom 186
|
nr 3
267-274
EN
It is proved that if $J_{i}$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_{i} = N_{i} + Q_{i}$, where $N_{i}$ is normal and $Q_{i}$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.
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