On a theorem of Gelfand and its local generalizations

• Autorzy: Driss Drissi
• Języki publikacji: EN

Abstrakty

EN

In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = {1}, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille's results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand's theorem for m commuting bounded operators, $T_1,..., T_m$, on a Banach space X.

Słowa kluczowe

EN

locally power-bounded operator; local spectrum; local spectral radius

Kategorie tematyczne

• 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)
• 47B10: Operators belonging to operator ideals (nuclear, p -summing, in the Schatten-von Neumann classes, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 123
• Numer: 2
• Strony: 185-194

Daty

wydano

1997

otrzymano

1996-02-20

poprawiono

1996-09-24

Twórcy

autor

Driss Drissi

• Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969 Safat 13060, Kuwait

Bibliografia

• [1] Allan G. R.: Sums of idempotents and a lemma of N. J. Kalton, Studia Math. 121 (1996), 185-192.
• [2] Allan G. R. and Ransford T. J.: Power-dominated elements in a Banach algebra, ibid. 94 (1989), 63-79.
• [3] Atzmon A.: Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144 (1980), 27-63.
• [4] Aupetit B. and Drissi D.: Some spectral inequalities involving generalized scalar operators, Studia Math. 109 (1994), 51-66.
• [5] Aupetit B. and Drissi D.: Local spectrum theory and subharmonicity, Proc. Edinburgh Math. Soc. 39 (1996), 571-579.
• [6] Batty C. J. K.: Asymptotic behaviour of semigroups of operators, in: Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 35-52.
• [7] Bernau S. J. and Huijsmans C. B.: On the positivity of the unit element in a normed lattice ordered algebra, Studia Math. 97 (1990), 143-149.
• [8] Boas R. P.: Entire Functions, Academic Press, New York, 1954.
• [9] Bohnenblust H. F. and Karlin S.: Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. 62 (1955), 217-229.
• [10] Brunel A. et Émilion R.: Sur les opérateurs positifs à moyennes bornées, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984),103-106.
• [11] Colojoară I. and Foiaş C.: Theory of Generalized Spectral Operators, Gordon and Breach, 1968.
• [12] Émilion R.: Mean-bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), 1-14.
• [13] Esterle J.: Quasimultipliers, representations of $H^∞$, and the closed ideal problem for commutative Banach algebras, in: Radical Banach Algebras and Automatic Continuity (Long Beach, Calif., 1981), Lecture Notes in Math. 975, Springer, 1983, 66-162.
• [14] Gelfand I.: Zur Theorie der Charaktere der Abelschen topologischen Gruppen, Mat. Sb. 9 (1941), 49-50.
• [15] Hille E.: On the theory of characters of groups and semi-groups in normed vector rings, Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 58-60.
• [16] Hille E. and Phillips R. S.: Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, 1957.
• [17] Katznelson Y. and Tzafriri L.: On power bounded operators, J. Funct. Anal. 68 (1986), 313-328.
• [18] Laursen K. B. and Mbekhta M.: Operators with finite chain length, Proc. Amer. Math. Soc. 123 (1995), 3443-3448.
• [19] Levin B. Ja.: Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, 1964.
• [20] Lumer G.: Spectral operators, hermitian operators, and bounded groups, Acta Sci. Math. (Szeged) 25 (1964), 75-85.
• [21] Lumer G.: Spectral operators, Bounded groups and a theorem of Gelfand, Rev. Un. Mat. Argentina 25 (1971), 239-245.
• [22] Lumer G. and Phillips R. S.: Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679-698.
• [23] Lyubich Yu. and Zemánek J.: Precompactness in the uniform ergodic theory, Studia Math. 112 (1994), 89-97.
• [24] Mbekhta M. and Vasilescu F.-H.: Uniformly ergodic multioperators, Trans. Amer. Math. Soc. 347 (1995), 1847-1854.
• [25] Mbekhta M. and Zemánek J.: Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1155-1158.
• [26] Pedersen T. V.: Norms of powers in Banach algebras, Bull. London Math. Soc. 27 (1995), 305-316.
• [27] Pytlik T.: Analytic semigroups in Banach algebras and a theorem of Hille, Colloq. Math. 51 (1987), 287-294.
• [28] Shilov G. E.: On a theorem of I. M. Gel'fand and its generalizations, Dokl. Akad. Nauk SSSR 72 (1950), 641-644 (in Russian).
• [29] Sinclair A. M.: The norm of a hermitian element in a Banach Algebra, Proc. Amer. Math. Soc. 28 (1971), 446-450.
• [30] J. G. Stampfli and J. P. Williams, Growth conditions and the numerical range in a Banach algebra, Tôhoku Math. J. 20 (1968), 417-424.
• [31] Vũ Quôc Phóng, A short proof of the Y. Katznelson's and L. Tzafriri's theorem, Proc. Amer. Math. Soc. 115 (1992), 1023-1024.
• [32] Zemánek J.: Sur les itérations des opérateurs, Publ. Math. Univ. Pierre et Marie Curie, Séminaire d'Initiation à l'Analyse, 1994.
• [33] Zemánek J.: Sur les itérations des opérateurs, On the Gelfand-Hille theorems, in: Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 369-385.