In search of the invisible spectrum
Annales de l'Institut Fourier, Volume 49 (1999) no. 6, p. 1925-1998
In this paper, we begin the study of the phenomenon of the “invisible spectrum” for commutative Banach algebras. Function algebras, formal power series and operator algebras will be considered. A quantitative treatment of the famous Wiener-Pitt-Sreider phenomenon for measure algebras on locally compact abelian (LCA) groups is given. Also, our approach includes efficient sharp estimates for resolvents and solutions of higher Bezout equations in terms of their spectral bounds. The smallest “spectral hull” of a given closed set is introduced and studied; it permits the definition of a uniformly bounded functional calculus. In this paper, the program traced above is realized for the following algebras: the measure algebras of LCA groups; the measure algebras of a large class of topological abelian semigroups; their subalgebras - the (semi)group algebra of LCA (semi)groups, the algebra of almost periodic functions, the algebra of absolutely convergent Dirichlet series. Upper and lower estimates for the best majorants and critical constants are obtained.
Le sujet principal de ce travail est l’aspect quantitatif du phénomène du “spectre invisible” dans les algèbres de Banach commutatives. De ce point de vue, nous étudions certaines algèbres de fonctions, les algèbres de séries entières formelles, ainsi que certaines algèbres d’opérateurs. Cela nous permet de donner une interprétation quantitative du célèbre effet de Wiener, Pitt et Sreider pour les algèbres convolutives des mesures sur les groupes abéliens localement compacts. L’approche développée permet de trouver des majorants explicites (et parfois exacts), dépendant uniquement des bornes inférieures spectrales, pour les résolvantes et des solutions des équations de Bezout d’ordres supérieurs. Nous utilisons ces résultats pour définir et calculer la “moindre enveloppe spectrale” d’un ensemble donné, ainsi que le calcul fonctionnel uniformément continu. Dans ce travail, ce programme est réalisé pour les algèbres suivantes : les algèbres des mesures sur des groupes et semi-groupes abéliens localement compacts; leurs sous-algèbres, comme l’algèbre L 1 (G) des fonctions presque périodiques, l’algèbre des séries de Dirichlet absolument convergentes, etc. Pour toutes ces algèbres, nous trouvons les meilleurs majorants pour les inverses, ainsi que les “constantes critiques” correspondantes.
     author = {Nikolski, Nikolai},
     title = {In search of the invisible spectrum},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     pages = {1925-1998},
     doi = {10.5802/aif.1743},
     zbl = {0947.46035},
     mrnumber = {2001a:46053},
     language = {en},
     url = {}
Nikolski, Nikolai. In search of the invisible spectrum. Annales de l'Institut Fourier, Volume 49 (1999) no. 6, pp. 1925-1998. doi : 10.5802/aif.1743.

[B] J.-E. Björk, On the spectral radius formula in Banach algebras, Pacific J. Math., 40, n°2 (1972), 279-284. | MR 46 #7901 | Zbl 0235.46077

[Co] P. Cohen, A note on constructive methods in Banach algebras, Proc. Amer. Math. Soc., 12, n°1 (1961), 159-164. | MR 23 #A1827 | Zbl 0097.10602

[D] E.M. Dyn'Kin, Theorems of Wiener-Lévy type and estimates for Wiener-Hopf operators, Matematicheskie Issledovania, Kishinev, I. Gohberg ed., 8:3 (29) (1973), 14-25 (Russian).

[ENZ] O. El-Fallah, N. Nikolski, M. Zarrabi, Resolvent estimates in Beurling-Sobolev algebras, Algebra i Analiz, 10, n°6 (1998) ; English transl. : St. Petersburg Math. J., 10:6 (1999). | Zbl 0938.46047

[EZ] O. El-Fallah, M. Zarrabi, Estimates for solutions of Bezout equations in Beurling algebras, to appear.

[Gam] T.W. Gamelin, Uniform Algebras and Jensen Measures, London Math. Soc. Lect. Notes Series 32, Cambridge Univ. Press, Cambridge. | MR 81a:46058 | Zbl 0418.46042

[Gar] J.B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. | MR 83g:30037 | Zbl 0469.30024

[GRS] I.M. Gelfand, Raikov D.A., G.E. Shilov, Commutative normed rings (Russian), Fizmatgiz, Moscow, 1960 ; English transl. : Chelsea, New York, 1964.

[GMG] C.C. Graham, O.C. Mcgehee, Essays in Commutative Harmonic Analysis, Springer, Heidelberg-New York, 1979. | MR 81d:43001 | Zbl 0439.43001

[HKKR] H. Helson, J.-P. Kahane, Y. Katznelson, W. Rudin, The functions which operate on Fourier transforms, Acta Math., 102 (1959), 135-157. | MR 22 #6980 | Zbl 0091.10902

[HR] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, Vol. I and II., Springer, Heidelberg-New York, 1963 and 1970.

[K1] J.-P. Kahane, Séries de Fourier absolument convergentes, Springer, Heidelberg-New York, 1970. | MR 43 #801 | Zbl 0195.07602

[K2] J.-P. Kahane, Sur le théorème de Beurling-Pollard, Math. Scand., 21 (1967), 71-79. | MR 40 #629 | Zbl 0191.14201

[L] N. Leblanc, Les fonctions qui opèrent dans certaines algèbres à poids, Math. Scand., 25 (1969), 190-194. | MR 42 #760 | Zbl 0191.14202

[New] D.J. Newman, A simple proof of Wiener's 1/f theorem, Proc. Amer. Math. Soc., 48 (1975), 264-265. | MR 51 #1255 | Zbl 0296.42017

[N1] N. Nikolski, Treatise on the shift operator, Springer, Heidelberg-New York, 1986.

[N2] N. Nikolski, Norm control of inverses in radical and operator Banach algebras, to appear.

[R] M. Rosenblum, A corona theorem for countably many functions, Integral Equat. Oper. Theory, 3, n°1 (1980), 125-137. | MR 81e:46034 | Zbl 0452.46032

[Ru1] W. Rudin, Fourier Analysis on Groups, Interscience Tract n°12, Wiley, New York, 1962. | MR 27 #2808 | Zbl 0107.09603

[Ru2] W. Rudin, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc., 7 (1956), 808-811. | MR 18,472c | Zbl 0073.29701

[S] F.A. Shamoyan, Applications of Dzhrbashyan integral representations to certain problems of analysis, Doklady AN SSSR, 261 (1981), n°3, 557-561 (Russian) ; English transl. : Soviet Math. Doklady 24:3 (1981), 563-567. | Zbl 0501.30030

[Sh1] H.S. Shapiro, A counterexample in harmonic analysis, in Approximation Theory, Banach Center Publications, Warsaw (submitted 1975), Vol. 4 (1979), 233-236. | MR 81c:46047 | Zbl 0342.43005

[Sh2] H.S. Shapiro, Bounding the norm of the inverse elements in the Banach algebra of absolutely convergent Taylor series, Abstracts of the Sixth Summer St. Petersburg Meeting in Mathematical Analysis, St. Petersburg, June 22-24, 1997.

[Sr] Yu.A. Sreider, The structure of maximal ideals in rings of measures with involution, Matem. Sbornik, 27 (69) (1950), 297-318 (Russian) ; English transl. : AMS Transl., 81 (1953), 28 pp.

[St] J.D. Stafney, An unbounded inverse property in the algebra of absolutely convergent Fourier series, Proc. Amer. Math. Soc., 18:1 (1967), 497-498. | MR 35 #4679 | Zbl 0158.05703

[T] J.L. Taylor, Measure algebras, CBMS Conf., n°16, Amer. Math. Soc., Providence, R.I., 1972.

[To1] V.A. Tolokonnikov, Estimates in Carleson's corona theorem and finitely generated ideals in the algebra H∞, Functional. Anal. i ego Prilozh, 14 (1980) 85-86 (Russian) ; English transl. : Funct. Anal. Appl., 14:4 (1980), 320-321. | MR 82a:46058 | Zbl 0457.46041

[To2] V.A. Tolokonnikov, Corona theorems in algebras of bounded analytic functions, Manuscript deposed in VINITI, Moscow, n° 251-84 DEP, 1984 (Russian).

[VP] S.A. Vinogradov, A.N. Petrov, The converse to the theorem on operation of analytic functions, and multiplicative properties of some subclasses of the Hardy space H∞ Zapiski Nauchnyh Semin. St. Petersburg Steklov Institute, 232 (1996), 50-72 (Russian). | Zbl 0895.30024

[Z] A. Zygmund, Trigonometric series, Vol. 1., Cambridge, The University Press, 1959. | Zbl 0085.05601