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1994 | 30 | 1 | 313-325
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Invariant subspaces and spectral mapping theorems

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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 More or less detailed proofs are given for results unpublished before or published in short communications; in some other cases we give a scheme of proof.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.
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  • Department of Mathematics, Vologda Polytechnical Institute, 15 Lenin St., 160008 Vologda, Russia
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  • [7] D. C. Kleinecke, On operator commutators, ibid. 8 (1957), 536-537.
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  • [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.
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  • [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).
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