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1998 | 128 | 2 | 159-169
Tytuł artykułu

Commutators of quasinilpotents and invariant subspaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
It is proved that the set Q of quasinilpotent elements in a Banach algebra is an ideal, i.e. equal to the Jacobson radical, if (and only if) the condition [Q,Q] ⊆ Q (or a similar condition concerning anticommutators) holds. In fact, if the inner derivation defined by a quasinilpotent element p maps Q into itself then p ∈ Rad A. Higher commutator conditions of quasinilpotents are also studied. It is shown that if a Banach algebra satisfies such a condition, then every quasinilpotent element has some fixed power in the Jacobson radical. These results are applied to topologically transitive representations. As a consequence, it is proved that a closed algebra of polynomially compact operators satisfying a higher commutator condition must have an invariant nest of closed subspaces, with "gaps" of bounded dimension. In particular, if [Q,Q] ⊆ Q, then the algebra must be triangularizable. An example is given showing that this may fail for more general algebras.
Słowa kluczowe
Czasopismo
Rocznik
Tom
128
Numer
2
Strony
159-169
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-02-03
poprawiono
1997-06-30
Twórcy
autor
  • Department of Mathematics, University of Athens, 15784 Athens, Greece
Bibliografia
  • [1] W. B. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433-532.
  • [2] B. Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Math. 735, Springer, 1979.
  • [3] B. Aupetit, A Primer on Spectral Theory, Springer, 1990.
  • [4] B. A. Barnes and A. Katavolos, Properties of quasinilpotents in some operator algebras, Proc. Roy. Irish Acad. Sect. A 93 (1993), 155-170.
  • [5] S. Grabiner, The nilpotency on Banach nil algebras, Proc. Amer. Math. Soc. 21 (1969), 510.
  • [6] D. Hadwin, E. Nordgren, M. Radjabalipour, H. Radjavi and P. Rosenthal, On simultaneous triangularization of collections of operators, Houston J. Math. 17 (1991), 581-602.
  • [7] A. Katavolos and H. Radjavi, Simultaneous triangularization of operators on a Banach space, J. London Math. Soc. (2) 41 (1990), 547-554.
  • [8] V. I. Lomonosov, Invariant subspaces for the family of operators which commute with a completely continuous operator, Funktsional. Anal. i Prilozhen. 7 (3) (1973) (in Russian); English transl.: Functional Anal. Appl. 7 (1973), 213-214.
  • [9] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211.
  • [10] M. Radjabalipour, Simultaneous triangularization of algebras of polynomially compact operators, Canad. Math. Bull. 34 (1991), 260-264.
  • [11] H. Radjavi, The Engel-Jacobson theorem revisited, J. Algebra 111 (1987), 427-430.
  • [12] H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer, 1973.
  • [13] J. R. Ringrose, Compact Nonselfadjoint Operators, Van Nostrand, New York, 1971.
  • [14] P. Rosenthal, Applications of Lomonosov's lemma to non-self-adjoint operator algebras, Proc. Roy. Irish Acad. Sect. A 74 (1974), 271-281.
  • [15] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge Univ. Press, 1976.
  • [16] Z. Słodkowski, W. Wojtyński and J. Zemánek, A note on quasinilpotent elements of a Banach algebra, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 131-134.
  • [17] E. Vesentini, On the subharmonicity of the spectral radius, Boll. Un. Mat. Ital. 4 (1968), 427-429.
  • [18] P. Vrbová, A remark concerning commutativity modulo the radical in Banach algebras, Comment. Math. Univ. Carolin. 22 (1981), 145-148.
  • [19] J. Zemánek, Spectral radius characterizations of commutativity in Banach algebras, Studia Math. 61 (1977), 257-268.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv128i2p159bwm
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