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In this paper, it is proved that the Banach algebra , generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and consists of polynomially compact operators. It is also proved that 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.
Peng Cao. "Quasinilpotent operators in operator Lie algebras II." Studia Mathematica 195.2 (2009): 193-200. <http://eudml.org/doc/285317>.
@article{PengCao2009, abstract = {In this paper, it is proved that the Banach algebra $\overline\{(ℒ)\}$, generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\overline\{(ℒ)\}$ consists of polynomially compact operators. It is also proved that $\overline\{(ℒ)\}$ 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.}, author = {Peng Cao}, journal = {Studia Mathematica}, keywords = {essentially nilpotent Lie algebra; quasinilpotent operator; compactly quasinilpotent algebra}, language = {eng}, number = {2}, pages = {193-200}, title = {Quasinilpotent operators in operator Lie algebras II}, url = {http://eudml.org/doc/285317}, volume = {195}, year = {2009}, }
TY - JOUR AU - Peng Cao TI - Quasinilpotent operators in operator Lie algebras II JO - Studia Mathematica PY - 2009 VL - 195 IS - 2 SP - 193 EP - 200 AB - In this paper, it is proved that the Banach algebra $\overline{(ℒ)}$, generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\overline{(ℒ)}$ consists of polynomially compact operators. It is also proved that $\overline{(ℒ)}$ 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. LA - eng KW - essentially nilpotent Lie algebra; quasinilpotent operator; compactly quasinilpotent algebra UR - http://eudml.org/doc/285317 ER -