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1994 | 30 | 1 | 53-100
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Continuity of spectrum and spectral radius in Banach algebras

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This survey deals with necessary and/or sufficient conditions for continuity of the spectrum and spectral radius functions at a point of a Banach algebra.
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  • Dipartimento di Matematica dell'Università di Genova, Via L. B. Alberti, 4, 16132 Genova, Italy
  • [AK] M. B. Abalovich and N. Y. Krupnik, A topology on the set of maximal ideals of a Banach PI-algebra, Amer. Math. Soc. Transl. (2) 142 (1989), 83-90.
  • [Ac1] S. T. M. Ackermans, On the principal extension of complex sets in a Banach algebra, Indag. Math. 29 (1967), 146-150.
  • [Ac2] S. T. M. Ackermans, A case of strong spectral continuity, ibid. 30 (1968), 455-459.
  • [Ap] C. Apostol, The spectrum and the spectral radius as functions in Banach algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 975-978.
  • [AFHV] C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu, Approximation of Hilbert Space Operators, Vol. II, Res. Notes Math. 102, Pitman, 1984.
  • [AM] C. Apostol and B. B. Morrel, On uniform approximation of operators by simple models, Indiana Univ. Math. J. 26 (1977), 427-442.
  • [Au1] B. Aupetit, Continuité du spectre dans les algèbres de Banach avec involution, Pacific J. Math. 56 (1975), 321-324.
  • [Au2] B. Aupetit, Caractérisation spectrale des algèbres de Banach commutatives, ibid. 63 (1976), 23-35.
  • [Au3] B. Aupetit, Continuité uniforme du spectre dans les algèbres de Banach avec involution, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), 1125-1127.
  • [Au4] B. Aupetit, Continuité et uniforme continuité du spectre dans les algèbres de Banach, Studia Math. 61 (1977), 99-114.
  • [Au5] B. Aupetit, La deuxième conjecture de Hirschfeld-Żelazko pour les algèbres de Banach est fausse, Proc. Amer. Math. Soc. 70 (1978), 161-162.
  • [Au6] B. Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Math. 735, Springer, 1979.
  • [BD] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb. 80, Springer, 1973.
  • [B1] L. Burlando, On two subsets of a Banach algebra that are related to the continuity of spectrum and spectral radius, Linear Algebra Appl. 84 (1986), 251-269.
  • [B2] L. Burlando, Two sets of continuity points of the spectrum and spectral radius functions in an algebra of operators, Istit. Lombardo Accad. Sci. Lett. Rend. A 120 (1986), 135-147 (1987).
  • [B3] L. Burlando, Continuity of spectrum and spectral radius in algebras of operators, Ann. Fac. Sci. Toulouse Math. (5) 9 (1988), 5-54.
  • [B4] L. Burlando, On the problem of invariance under holomorphic functions for a set of continuity points of the spectrum function, Czechoslovak Math. J. 39 (1989), 95-110.
  • [B5] L. Burlando, On continuity of the spectral radius function in Banach algebras, Ann. Mat. Pura Appl. (4) 156 (1990), 357-380.
  • [B6] L. Burlando, On continuity of the spectrum function in Banach algebras, Riv. Mat. Pura Appl. 8 (1991), 131-152.
  • [B7] L. Burlando, On continuity of the spectrum function in Banach algebras with good ideal structure, ibid. 9 (1991), 7-21.
  • [B8] L. Burlando, Spectral continuity, Atti Sem. Mat. Fis. Univ. Modena 40 (1992), 591-605.
  • [B9] L. Burlando, Spectral continuity in some Banach algebras, Rocky Mountain J. Math. 23 (1993), 17-39.
  • [B10] L. Burlando, Banach algebras on which the spectrum function is continuous, to appear.
  • [Ca] S. R. Caradus, Generalized Inverses and Operator Theory, Queen's Papers in Pure and Appl. Math. 50, Queen's University, Kingston, Ont., 1978.
  • [CPY] S. R. Caradus, W. E. Pfaffenberger and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, Lecture Notes in Pure Appl. Math. 9, Marcel Dekker, 1974.
  • [Cl] J. M. Clauss, Elementary chains of invariant subspaces of a Banach space, preprint.
  • [Co] J. B. Conway, On the Calkin algebra and the covering homotopy property, Trans. Amer. Math. Soc. 211 (1975), 135-142.
  • [CM1] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integral Equations Operator Theory 2 (1979), 174-198.
  • [CM2] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, II, ibid. 4 (1981), 459-503.
  • [CM3] J. B. Conway and B. B. Morrel, Behaviour of the spectrum under small perturbations, Proc. Roy. Irish Acad. Sect. A 81 (1981), 55-63.
  • [CM4] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, III, Integral Equations Operator Theory 6 (1983), 319-344.
  • [D] Z. Daoultzi-Malamou, Strong spectral continuity in topological matrix algebras, Boll. Un. Mat. Ital. A (7) 2 (1988), 213-219.
  • [GKre] I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. Math. Soc. Transl. (2) 13 (1960), 185-264.
  • [GKru] I. C. Gohberg and N. Y. Krupnik, Extension theorems for invertibility symbols in Banach algebras, Integral Equations Operator Theory 15 (1992), 991-1010.
  • [G] M. Gonzalez, A perturbation result for generalised Fredholm operators in the boundary of the group of invertible operators, Proc. Roy. Irish Acad. Sect. A 86 (1986), 123-126.
  • [GM] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874.
  • [Ha] P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, 1967.
  • [He1] D. A. Herrero, Continuity of spectral functions and the lakes of Wada, Pacific J. Math. 113 (1984), 365-371.
  • [He2] D. A. Herrero, Similarity-invariant continuous functions on ℒ(ℋ), Proc. Amer. Math. Soc. 97 (1986), 75-78.
  • [HS] D. A. Herrero and N. Salinas, Operators with disconnected spectra are dense, Bull. Amer. Math. Soc. 78 (1972), 525-526.
  • [HY] J. G. Hocking and G. S. Young, Topology, Addison-Wesley, 1961.
  • [J] J. Janas, Note on the spectrum and joint spectrum of hyponormal and Toeplitz operators, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 957-961.
  • [Ka1] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322.
  • [Ka2] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, 1966.
  • [Kr] N. Y. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, Oper. Theory: Adv. Appl. 26, Birkhäuser, 1987.
  • [LS] S. Levi and Z. Słodkowski, Measurability properties of spectra, Proc. Amer. Math. Soc. 98 (1986), 225-231.
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I. Sequence Spaces, Ergeb. Math. Grenzgeb. 97, Springer, 1977.
  • [L] E. Luft, The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslovak Math. J. 18 (1968), 595-605.
  • [Mi] B. S. Mityagin, The homotopy structure of the linear group of a Banach space, Russian Math. Surveys 25(5) (1970), 59-103.
  • [Mu] G. J. Murphy, Continuity of the spectrum and spectral radius, Proc. Amer. Math. Soc. 82 (1981), 619-621.
  • [N] J. D. Newburgh, The variation of spectra, Duke Math. J. 18 (1951), 165-176.
  • [P] A. Pietsch, Operator Ideals, North-Holland Math. Library 20, North-Holland, 1980.
  • [PZ1] V. Pták and J. Zemánek, Continuité lipschitzienne du spectre comme fonction d'un opérateur normal, Comment. Math. Univ. Carolin. 17 (1976), 507-512.
  • [PZ2] V. Pták and J. Zemánek, On uniform continuity of the spectral radius in Banach algebras, Manuscripta Math. 20 (1977), 177-189.
  • [Q] C. Qiu, Continuity of spectral functions of operators, Chinese Ann. Math. Ser. A 10 (1989), 621-627 (in Chinese).
  • [R] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, 1960.
  • [TL] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., Wiley, 1980.
  • [Ze1] J. Zemánek, Spectral radius characterizations of commutativity in Banach algebras, Studia Math. 61 (1977), 257-268.
  • [Ze2] J. Zemánek, Spectral characterization of two-sided ideals in Banach algebras, ibid. 67 (1980), 1-12.
  • [Ze3] J. Zemánek, An analytic Laffey-West decomposition, Proc. Roy. Irish Acad. Sect. A 92 (1992), 101-106.
  • [Zh] W. Q. Zhang, Continuity of set-valued mappings and some applications to the continuity of spectra of operators, J. Math. (Wuhan) 7 (1987), 285-290 (in Chinese).
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