PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
1994 | 109 | 1 | 51-66
Tytuł artykułu

Some spectral inequalities involving generalized scalar operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In 1971, Allan Sinclair proved that for a hermitian element h of a Banach algebra and λ complex we have ∥λ + h∥ = r(λ + h), where r denotes the spectral radius. Using Levin's subordination theory for entire functions of exponential type, we extend this result locally to a much larger class of generalized spectral operators. This fundamental result improves many earlier results due to Gelfand, Hille, Colojoară-Foiaş, Vidav, Dowson, Dowson-Gillespie-Spain, Crabb-Spain, I. & V. Istrăţescu, Barnes, Pytlik, Boyadzhiev and others.
Słowa kluczowe
Twórcy
autor
  • Département de Mathématiques et de Statistique, Université Laval, Québec, Qué., Canada G1K 7P4
autor
  • Département de Mathématiques et de Statistique, Université Laval, Québec, Qué., Canada G1K 7P4
Bibliografia
  • [1] G. R. Allan and T. J. Ransford, Power-dominated elements in a Banach algebra, Studia Math. 94 (1989), 63-79.
  • [2] C. Apostol, Teorie spectrală şi calcul functional, Stud. Cerc. Mat. 20 (1968), 635-668.
  • [3] B. Aupetit, A Primer on Spectral Theory, Springer, 1991.
  • [4] B. Aupetit and D. Drissi, Local spectrum theory revisited, to appear.
  • [5] B. A. Barnes, Operators which satisfy polynomial growth conditions, Pacific J. Math. 138 (1987), 209-219.
  • [6] R. G. Bartle and C. A. Kariotis, Some localizations of the spectral mapping theorem, Duke Math. J. 40 (1973), 651-660.
  • [7] R. P. Boas, Entire Functions, Academic Press, 1954.
  • [8] B. Bollobás, A property of hermitian elements, J. London Math. Soc. 4 (1971), 379-380.
  • [9] F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Ser. 2, Cambridge University Press, 1971.
  • [10] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Ser. 10, Cambridge University Press, 1973.
  • [11] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973.
  • [12] H. N. Bojadjiev [K. N. Boyadzhiev], New applications of Bernstein inequality to the theory of operators: a local Sinclair lemma and a generalization of the Fuglede-Putnam theorem, in: Complex Analysis and Applications 85, Sofia, 1986, 97-104.
  • [13] H. N. Bojadjiev [K. N. Boyadzhiev], Sinclair type inequalities for the local spectral radius and related topics, Israel J. Math. 57 (1987), 272-284.
  • [14] A. Browder, On Bernstein's inequality and the norm of hermitian operators, Amer. Math. Monthly 78 (1971), 871-873.
  • [15] I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon and Breach, 1968.
  • [16] M. J. Crabb and P. G. Spain, Commutators and normal operators, Glasgow Math. J. 18 (1977), 197-198.
  • [17] H. R. Dowson, Some properties of prespectral operators, Proc. Roy. Irish Acad. 74 (1974), 207-221.
  • [18] H. R. Dowson, T. A. Gillespie and P. G. Spain, A commutativity theorem for hermitian operators, Math. Ann. 220 (1976), 215-217.
  • [19] D. Drissi, Quelques inégalités spectrales pour les opérateurs scalaires généralisés, Ph.D. thesis, Université Laval, 1993.
  • [20] I. Erdelyi and R. Lange, Spectral Decompositions on Banach Spaces, Lecture Notes in Math. 623, Springer, 1977.
  • [21] C. Foiaş, Une application des distributions vectorielles à la théorie spectrale, Bull. Sci. Math. 84 (1960), 147-158.
  • [22] C. K. Fong, Normal operators on Banach spaces, Glasgow Math. J. 20 (1979), 163-168.
  • [23] I. Gelfand, Zur theorie der Charaktere der abelschen topologischen Gruppen, Rec. Math. N.S. (Mat. Sb.) 9 (51) (1941), 49-50.
  • [24] E. Hille, On the theory of characters of groups and semi-groups in normed vector rings, Proc. Nat. Acad. Sci. 30 (1944), 58-60.
  • [25] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, 1957.
  • [26] I. Istrăţescu and V. Istrăţescu, A note on the Weyl's spectrum of an operator, Rev. Roumaine Math. Pures Appl. 15 (1970), 1445-1447.
  • [27] V. È. Kacnel'son [V. È. Katsnel'son], A conservative operator has norm equal to its spectral radius, Mat. Issled. 5 (3) (17) (1970), 186-189 (in Russian).
  • [28] S. Kantorovitz, Classification of operators by means of their operational calculus, Trans. Amer. Math. Soc. 115 (1965), 194-224.
  • [29] G. K. Leaf, A spectral theory for a class of linear operators, Pacific J. Math. 13 (1963), 141-155.
  • [30] B. Ja. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., 1964.
  • [31] T. Pytlik, Analytic semigroups in Banach algebras and a theorem of Hille, Colloq. Math. 51 (1987), 287-294.
  • [32] F. Riesz et B. Sz.-Nagy, Leçons d'analyse fonctionnelle, Acad. Sci. Hongrie, Szeged, 1955.
  • [33] W. Rudin, Functional Analysis, McGraw-Hill, 1973.
  • [34] A. M. Sinclair, The norm of a hermitian element in a Banach algebra, Proc. Amer. Math. Soc. 28 (1971), 446-450.
  • [35] D. R. Smart, Conditionally convergent expansions, J. Austral. Math. Soc. 1 (1960), 319-333.
  • [36] B. G. Tillman, Vector-valued distributions and the spectral theorem for self-adjoint operators in Hilbert space, Bull. Amer. Math. Soc. 69 (1963), 67-71.
  • [37] I. Vidav, Eine metrische Kennzeichnung der selbstadjungierten Operatoren, Math. Z. 66 (1956), 121-128.
  • [38] P. Vrbová, On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23 (1973), 483-492.
  • [39] K. K. Warner, A note on a theorem of Weyl, Proc. Amer. Math. Soc. 23 (1969), 469-471.
  • [40] F. Wolf, Operators in Banach space which admit a generalized spectral decomposition, Nederl. Akad. Wetensch. Indag. Math. 19 (1957), 302-311.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv109i1p51bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.