In search of the invisible spectrum

Nikolai Nikolski

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 6, page 1925-1998
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Nikolski, Nikolai. "In search of the invisible spectrum." Annales de l'institut Fourier 49.6 (1999): 1925-1998. <http://eudml.org/doc/75407>.

@article{Nikolski1999,
abstract = {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.},
author = {Nikolski, Nikolai},
journal = {Annales de l'institut Fourier},
keywords = {invisible spectrum; LCA groups; measure algebra; Wiener-Pitt-Sreider phenomenon; norm controlled inversion; cyclic groups; spectral hulls; function algebras; norm-controlled functional calculi; commutative Banach algebras; formal power series; operator algebras; Bézout equations; algebra of almost periodic functions; algebra of absolutely convergent Dirichlet series; best majorants; critical constants},
language = {eng},
number = {6},
pages = {1925-1998},
publisher = {Association des Annales de l'Institut Fourier},
title = {In search of the invisible spectrum},
url = {http://eudml.org/doc/75407},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Nikolski, Nikolai
TI - In search of the invisible spectrum
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 6
SP - 1925
EP - 1998
AB - 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.
LA - eng
KW - invisible spectrum; LCA groups; measure algebra; Wiener-Pitt-Sreider phenomenon; norm controlled inversion; cyclic groups; spectral hulls; function algebras; norm-controlled functional calculi; commutative Banach algebras; formal power series; operator algebras; Bézout equations; algebra of almost periodic functions; algebra of absolutely convergent Dirichlet series; best majorants; critical constants
UR - http://eudml.org/doc/75407
ER -

References

top
  1. [B] J.-E. BJÖRK, On the spectral radius formula in Banach algebras, Pacific J. Math., 40, n°2 (1972), 279-284. Zbl0235.46077MR46 #7901
  2. [Co] P. COHEN, A note on constructive methods in Banach algebras, Proc. Amer. Math. Soc., 12, n°1 (1961), 159-164. Zbl0097.10602MR23 #A1827
  3. [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). 
  4. [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). Zbl0938.46047
  5. [EZ] O. EL-FALLAH, M. ZARRABI, Estimates for solutions of Bezout equations in Beurling algebras, to appear. Zbl1089.46030
  6. [Gam] T.W. GAMELIN, Uniform Algebras and Jensen Measures, London Math. Soc. Lect. Notes Series 32, Cambridge Univ. Press, Cambridge. Zbl0418.46042MR81a:46058
  7. [Gar] J.B. GARNETT, Bounded analytic functions, Academic Press, New York, 1981. Zbl0469.30024MR83g:30037
  8. [GRS] I.M. GELFAND, RAIKOV D.A., G.E. SHILOV, Commutative normed rings (Russian), Fizmatgiz, Moscow, 1960 ; English transl. : Chelsea, New York, 1964. 
  9. [GMG] C.C. GRAHAM, O.C. MCGEHEE, Essays in Commutative Harmonic Analysis, Springer, Heidelberg-New York, 1979. Zbl0439.43001MR81d:43001
  10. [HKKR] H. HELSON, J.-P. KAHANE, Y. KATZNELSON, W. RUDIN, The functions which operate on Fourier transforms, Acta Math., 102 (1959), 135-157. Zbl0091.10902MR22 #6980
  11. [HR] E. HEWITT, K.A. ROSS, Abstract Harmonic Analysis, Vol. I and II., Springer, Heidelberg-New York, 1963 and 1970. 
  12. [K1] J.-P. KAHANE, Séries de Fourier absolument convergentes, Springer, Heidelberg-New York, 1970. Zbl0195.07602MR43 #801
  13. [K2] J.-P. KAHANE, Sur le théorème de Beurling-Pollard, Math. Scand., 21 (1967), 71-79. Zbl0191.14201MR40 #629
  14. [L] N. LEBLANC, Les fonctions qui opèrent dans certaines algèbres à poids, Math. Scand., 25 (1969), 190-194. Zbl0191.14202MR42 #760
  15. [New] D.J. NEWMAN, A simple proof of Wiener's 1/f theorem, Proc. Amer. Math. Soc., 48 (1975), 264-265. Zbl0296.42017MR51 #1255
  16. [N1] N. NIKOLSKI, Treatise on the shift operator, Springer, Heidelberg-New York, 1986. 
  17. [N2] N. NIKOLSKI, Norm control of inverses in radical and operator Banach algebras, to appear. 
  18. [R] M. ROSENBLUM, A corona theorem for countably many functions, Integral Equat. Oper. Theory, 3, n°1 (1980), 125-137. Zbl0452.46032MR81e:46034
  19. [Ru1] W. RUDIN, Fourier Analysis on Groups, Interscience Tract n°12, Wiley, New York, 1962. Zbl0107.09603MR27 #2808
  20. [Ru2] W. RUDIN, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc., 7 (1956), 808-811. Zbl0073.29701MR18,472c
  21. [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. Zbl0501.30030
  22. [Sh1] H.S. SHAPIRO, A counterexample in harmonic analysis, in Approximation Theory, Banach Center Publications, Warsaw (submitted 1975), Vol. 4 (1979), 233-236. Zbl0342.43005MR81c:46047
  23. [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. 
  24. [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. 
  25. [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. Zbl0158.05703MR35 #4679
  26. [T] J.L. TAYLOR, Measure algebras, CBMS Conf., n°16, Amer. Math. Soc., Providence, R.I., 1972. 
  27. [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. Zbl0457.46041MR82a:46058
  28. [To2] V.A. TOLOKONNIKOV, Corona theorems in algebras of bounded analytic functions, Manuscript deposed in VINITI, Moscow, n° 251-84 DEP, 1984 (Russian). 
  29. [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). Zbl0895.30024
  30. [Z] A. ZYGMUND, Trigonometric series, Vol. 1., Cambridge, The University Press, 1959. Zbl0085.05601

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.