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## Banach Center Publications

1994 | 30 | 1 | 313-325
Tytuł artykułu

### Invariant subspaces and spectral mapping theorems

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We discuss some results and problems connected with estimation of spectra of operators (or elements of general Banach algebras) which are expressed as polynomials in several operators, noncommuting but satisfying weaker conditions of commutativity type (for example, generating a nilpotent Lie algebra). These results have applications in the theory of invariant subspaces; in fact, such applications were the motivation for consideration of spectral problems. More or less detailed proofs are given for results unpublished before or published in short communications; in some other cases we give a scheme of proof. The author is obliged to J. A. Erdos, V. S. Guba and especially to Yu. V. Turovskiĭ for useful discussions.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
313-325
Opis fizyczny
Daty
wydano
1994
Twórcy
autor
• Department of Mathematics, Vologda Polytechnical Institute, 15 Lenin St., 160008 Vologda, Russia
Bibliografia
• [1] J. A. Erdos, The commutant of a Volterra operator, Integral Equations Operator Theory 5 (1982), 127-130.
• [2] A. S. Faĭnshteĭn, Joint spectrum of Taylor type for families of operators generating nilpotent Lie algebras, preprint, 1989 (in Russian).
• [3] Ş. Frunză, Generalized weights for operator Lie algebras, in: Spectral Theory, Banach Center Publ. 8, PWN, Warszawa, 1982, 281-287.
• [4] V. S. Guba, An associative algebra with one relation of Engel type, preprint, 1990 (in Russian).
• [5] R. Harte, Spectral mapping theorems, Proc. Roy. Irish Acad. 72A (1972), 89-107.
• [6] E. V. Kissin, Invariant subspaces for derivations, Proc. Amer. Math. Soc. 102 (1988), 95-101.
• [7] D. C. Kleinecke, On operator commutators, ibid. 8 (1957), 536-537.
• [8] V. I. Lomonosov, On invariant subspaces of a family of operators commuting with a completely continuous operator, Funktsional. Anal. i Prilozhen. 7 (3) (1973), 55-56 (in Russian).
• [9] G. J. Murphy, Triangularizable algebras of compact operators, Proc. Amer. Math. Soc. 84 (1982), 354-356.
• [10] Yu. S. Samoĭlenko and V. S. Shul'man, On representations of relations i[A,B]=f(A)+g(B), Ukrainian Math. J. 43 (1991), 110-114.
• [11] D. Sarason, Generalized interpolation in $H^∞$, Trans. Amer. Math. Soc. 127 (1967), 179-203.
• [12] F. V. Shirokov, The proof of Kaplansky's hypothesis, Uspekhi Mat. Nauk 11 (4) (1956), 167-168 (in Russian).
• [13] V. S. Shul'man, On transitivity of some operator spaces, Funktsional. Anal. i Prilozhen. 16 (1) (1982), 91-92 (in Russian).
• [14] V. S. Shul'man, On invariant subspaces of compact operators, ibid. 18 (2) (1984), 85-86 (in Russian).
• [15] Z. Słodkowski and W. Żelazko, On joint spectra of commuting systems of linear operators, Studia Math. 50 (1974), 127-148.
• [16] J. L. Taylor, A joint spectrum of several commuting operators, J. Funct. Anal. 6 (1970), 172-191.
• [17] J. L. Taylor, A general framework for a multioperator functional calculus, Adv. in Math. 9 (1972), 183-252.
• [18] Yu. V. Turovskiĭ, The mapping of the Harte spectrum by polynomials for n-commutative families of elements of a Banach algebra, in: Spectral Theory of Operators and its Applications, No. 5, Elm, Baku, 1984, 152-177 (in Russian).
• [19] Yu. V. Turovskiĭ, On spectral properties of some Lie subalgebras and the spectral radius of subsets of a Banach algebra, in: Spectral Theory of Operators and its Applications, No. 6, Elm, Baku, 1985, 144-181 (in Russian).
• [20] Yu. V. Turovskiĭ, On commutativity modulo the Jacobson radical of the associative envelope of a Lie algebra, in: Spectral Theory of Operators and its Applications, No. 8, Elm, Baku, 1987, 199-211 (in Russian).
• [21] L. L. Vaksman and D. L. Gurariĭ, On algebras containing compact operators, Funktsional Anal. i Prilozhen. 8 (4) (1974), 81-82 (in Russian).
Typ dokumentu
Bibliografia
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