Invariant subspaces and spectral mapping theorems

V. Shul'man

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 313-325
  • ISSN: 0137-6934

Abstract

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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 More or less detailed proofs are given for results unpublished before or published in short communications; in some other cases we give a scheme of proof.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.

How to cite

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Shul'man, V.. "Invariant subspaces and spectral mapping theorems." Banach Center Publications 30.1 (1994): 313-325. <http://eudml.org/doc/262761>.

@article{Shulman1994,
abstract = {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 More or less detailed proofs are given for results unpublished before or published in short communications; in some other cases we give a scheme of proof.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.},
author = {Shul'man, V.},
journal = {Banach Center Publications},
keywords = {spectra of operators; invariant subspaces; spectral problems},
language = {eng},
number = {1},
pages = {313-325},
title = {Invariant subspaces and spectral mapping theorems},
url = {http://eudml.org/doc/262761},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Shul'man, V.
TI - Invariant subspaces and spectral mapping theorems
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 313
EP - 325
AB - 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.
LA - eng
KW - spectra of operators; invariant subspaces; spectral problems
UR - http://eudml.org/doc/262761
ER -

References

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  1. [1] J. A. Erdos, The commutant of a Volterra operator, Integral Equations Operator Theory 5 (1982), 127-130. Zbl0473.47015
  2. [2] A. S. Faĭnshteĭn, Joint spectrum of Taylor type for families of operators generating nilpotent Lie algebras, preprint, 1989 (in Russian). Zbl1005.47008
  3. [3] Ş. Frunză, Generalized weights for operator Lie algebras, in: Spectral Theory, Banach Center Publ. 8, PWN, Warszawa, 1982, 281-287. 
  4. [4] V. S. Guba, An associative algebra with one relation of Engel type, preprint, 1990 (in Russian). 
  5. [5] R. Harte, Spectral mapping theorems, Proc. Roy. Irish Acad. 72A (1972), 89-107. Zbl0206.13301
  6. [6] E. V. Kissin, Invariant subspaces for derivations, Proc. Amer. Math. Soc. 102 (1988), 95-101. Zbl0647.47006
  7. [7] D. C. Kleinecke, On operator commutators, ibid. 8 (1957), 536-537. Zbl0079.12904
  8. [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. [9] G. J. Murphy, Triangularizable algebras of compact operators, Proc. Amer. Math. Soc. 84 (1982), 354-356. Zbl0492.47023
  10. [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. [11] D. Sarason, Generalized interpolation in H , Trans. Amer. Math. Soc. 127 (1967), 179-203. Zbl0145.39303
  12. [12] F. V. Shirokov, The proof of Kaplansky's hypothesis, Uspekhi Mat. Nauk 11 (4) (1956), 167-168 (in Russian). 
  13. [13] V. S. Shul'man, On transitivity of some operator spaces, Funktsional. Anal. i Prilozhen. 16 (1) (1982), 91-92 (in Russian). 
  14. [14] V. S. Shul'man, On invariant subspaces of compact operators, ibid. 18 (2) (1984), 85-86 (in Russian). 
  15. [15] Z. Słodkowski and W. Żelazko, On joint spectra of commuting systems of linear operators, Studia Math. 50 (1974), 127-148. Zbl0306.47014
  16. [16] J. L. Taylor, A joint spectrum of several commuting operators, J. Funct. Anal. 6 (1970), 172-191. Zbl0233.47024
  17. [17] J. L. Taylor, A general framework for a multioperator functional calculus, Adv. in Math. 9 (1972), 183-252. Zbl0271.46041
  18. [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. [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. [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. [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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