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.
@article{bwmeta1.element.bwnjournal-article-bcpv30z1p53bwm,
author = {Burlando, Laura},
title = {Continuity of spectrum and spectral radius in Banach algebras},
journal = {Banach Center Publications},
volume = {29},
year = {1994},
pages = {53-100},
zbl = {0802.46062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv30z1p53bwm}
}
Burlando, Laura. Continuity of spectrum and spectral radius in Banach algebras. Banach Center Publications, Tome 29 (1994) pp. 53-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv30z1p53bwm/
[000] [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.
[001] [Ac1] S. T. M. Ackermans, On the principal extension of complex sets in a Banach algebra, Indag. Math. 29 (1967), 146-150. | Zbl 0147.33603
[002] [Ac2] S. T. M. Ackermans, A case of strong spectral continuity, ibid. 30 (1968), 455-459.
[003] [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. | Zbl 0396.46045
[004] [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. | Zbl 0572.47001
[005] [AM] C. Apostol and B. B. Morrel, On uniform approximation of operators by simple models, Indiana Univ. Math. J. 26 (1977), 427-442. | Zbl 0326.47012
[006] [Au1] B. Aupetit, Continuité du spectre dans les algèbres de Banach avec involution, Pacific J. Math. 56 (1975), 321-324. | Zbl 0306.46072
[007] [Au2] B. Aupetit, Caractérisation spectrale des algèbres de Banach commutatives, ibid. 63 (1976), 23-35.
[008] [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. | Zbl 0361.46043
[009] [Au4] B. Aupetit, Continuité et uniforme continuité du spectre dans les algèbres de Banach, Studia Math. 61 (1977), 99-114. | Zbl 0372.46044
[010] [Au5] B. Aupetit, La deuxième conjecture de Hirschfeld-Żelazko pour les algèbres de Banach est fausse, Proc. Amer. Math. Soc. 70 (1978), 161-162. | Zbl 0384.46030
[011] [Au6] B. Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Math. 735, Springer, 1979. | Zbl 0409.46054
[012] [BD] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb. 80, Springer, 1973.
[013] [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. | Zbl 0612.46046
[014] [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).
[015] [B3] L. Burlando, Continuity of spectrum and spectral radius in algebras of operators, Ann. Fac. Sci. Toulouse Math. (5) 9 (1988), 5-54. | Zbl 0618.47003
[016] [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. | Zbl 0819.47003
[017] [B5] L. Burlando, On continuity of the spectral radius function in Banach algebras, Ann. Mat. Pura Appl. (4) 156 (1990), 357-380. | Zbl 0726.46027
[018] [B6] L. Burlando, On continuity of the spectrum function in Banach algebras, Riv. Mat. Pura Appl. 8 (1991), 131-152. | Zbl 0764.46044
[019] [B7] L. Burlando, On continuity of the spectrum function in Banach algebras with good ideal structure, ibid. 9 (1991), 7-21. | Zbl 0762.46044
[020] [B8] L. Burlando, Spectral continuity, Atti Sem. Mat. Fis. Univ. Modena 40 (1992), 591-605. | Zbl 0792.47002
[021] [B9] L. Burlando, Spectral continuity in some Banach algebras, Rocky Mountain J. Math. 23 (1993), 17-39. | Zbl 0796.46035
[022] [B10] L. Burlando, Banach algebras on which the spectrum function is continuous, to appear.
[023] [Ca] S. R. Caradus, Generalized Inverses and Operator Theory, Queen's Papers in Pure and Appl. Math. 50, Queen's University, Kingston, Ont., 1978.
[024] [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. | Zbl 0299.46062
[025] [Cl] J. M. Clauss, Elementary chains of invariant subspaces of a Banach space, preprint.
[026] [Co] J. B. Conway, On the Calkin algebra and the covering homotopy property, Trans. Amer. Math. Soc. 211 (1975), 135-142. | Zbl 0281.46055
[027] [CM1] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integral Equations Operator Theory 2 (1979), 174-198. | Zbl 0419.47001
[028] [CM2] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, II, ibid. 4 (1981), 459-503. | Zbl 0468.47001
[029] [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. | Zbl 0493.47010
[030] [CM4] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, III, Integral Equations Operator Theory 6 (1983), 319-344.
[031] [D] Z. Daoultzi-Malamou, Strong spectral continuity in topological matrix algebras, Boll. Un. Mat. Ital. A (7) 2 (1988), 213-219. | Zbl 0652.46034
[032] [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.
[033] [GKru] I. C. Gohberg and N. Y. Krupnik, Extension theorems for invertibility symbols in Banach algebras, Integral Equations Operator Theory 15 (1992), 991-1010. | Zbl 0795.46043
[034] [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. | Zbl 0594.47010
[035] [GM] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. | Zbl 0827.46008
[036] [Ha] P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, 1967.
[037] [He1] D. A. Herrero, Continuity of spectral functions and the lakes of Wada, Pacific J. Math. 113 (1984), 365-371. | Zbl 0561.47002
[038] [He2] D. A. Herrero, Similarity-invariant continuous functions on ℒ(ℋ), Proc. Amer. Math. Soc. 97 (1986), 75-78. | Zbl 0593.47013
[039] [HS] D. A. Herrero and N. Salinas, Operators with disconnected spectra are dense, Bull. Amer. Math. Soc. 78 (1972), 525-526. | Zbl 0261.47005
[040] [HY] J. G. Hocking and G. S. Young, Topology, Addison-Wesley, 1961.
[041] [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. | Zbl 0317.47015
[042] [Ka1] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. | Zbl 0090.09003
[043] [Ka2] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, 1966.
[044] [Kr] N. Y. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, Oper. Theory: Adv. Appl. 26, Birkhäuser, 1987.
[045] [LS] S. Levi and Z. Słodkowski, Measurability properties of spectra, Proc. Amer. Math. Soc. 98 (1986), 225-231. | Zbl 0616.46046
[046] [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I. Sequence Spaces, Ergeb. Math. Grenzgeb. 97, Springer, 1977. | Zbl 0362.46013
[047] [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. | Zbl 0197.11301
[048] [Mi] B. S. Mityagin, The homotopy structure of the linear group of a Banach space, Russian Math. Surveys 25(5) (1970), 59-103. | Zbl 0232.47046
[049] [Mu] G. J. Murphy, Continuity of the spectrum and spectral radius, Proc. Amer. Math. Soc. 82 (1981), 619-621. | Zbl 0474.46038
[050] [N] J. D. Newburgh, The variation of spectra, Duke Math. J. 18 (1951), 165-176. | Zbl 0042.12302
[051] [P] A. Pietsch, Operator Ideals, North-Holland Math. Library 20, North-Holland, 1980.
[052] [PZ1] V. Pták and J. Zemánek, Continuité lipschitzienne du spectre comme fonction d'un opérateur normal, Comment. Math. Univ. Carolin. 17 (1976), 507-512. | Zbl 0341.47019
[053] [PZ2] V. Pták and J. Zemánek, On uniform continuity of the spectral radius in Banach algebras, Manuscripta Math. 20 (1977), 177-189. | Zbl 0347.46051
[054] [Q] C. Qiu, Continuity of spectral functions of operators, Chinese Ann. Math. Ser. A 10 (1989), 621-627 (in Chinese). | Zbl 0764.47004
[055] [R] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, 1960. | Zbl 0095.09702
[056] [TL] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., Wiley, 1980.
[057] [Ze1] J. Zemánek, Spectral radius characterizations of commutativity in Banach algebras, Studia Math. 61 (1977), 257-268. | Zbl 0321.46037
[058] [Ze2] J. Zemánek, Spectral characterization of two-sided ideals in Banach algebras, ibid. 67 (1980), 1-12.
[059] [Ze3] J. Zemánek, An analytic Laffey-West decomposition, Proc. Roy. Irish Acad. Sect. A 92 (1992), 101-106. | Zbl 0785.47008
[060] [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). | Zbl 0667.47003