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Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

Authors: Karlheinz Gröchenig and Michael Leinert
Journal: Trans. Amer. Math. Soc. 358 (2006), 2695-2711
MSC (2000): Primary 47B37, 47A60, 46H30, 42C15
Published electronically: January 24, 2006
MathSciNet review: 2204052
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Abstract: We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.

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  • 1. N. Atreas, J. J. Benedetto, and C. Karinakas.
    Local sampling for regular wavelet and Gabor expansions.
    Sampling Th. Signal Image Proc., 2(1):1-24, 2003. MR 2002854 (2004k:42050)
  • 2. R. Balan, P. Casazza, C. Heil, and Z. Landau.
    Density, redundancy, and localization of frames. II.
    Preprint, 2005.
  • 3. B. A. Barnes.
    The spectrum of integral operators on Lebesgue spaces.
    J. Operator Theory, 18(1):115-132, 1987. MR 0912815 (89i:46065)
  • 4. B. A. Barnes.
    Symmetric Banach $ \ast$-algebras: invariance of spectrum.
    Studia Math., 141(3):251-261, 2000. MR 1784672 (2001h:46098)
  • 5. H. Bölcskei and A. J. E. M. Janssen.
    Gabor frames, unimodularity, and window decay.
    J. Fourier Anal. Appl., 6(3):255-276, 2000. MR 1755143 (2001b:42044)
  • 6. L. H. Brandenburg.
    On identifying the maximal ideals in Banach algebras.
    J. Math. Anal. Appl., 50:489-510, 1975. MR 0377523 (51:13695)
  • 7. P. G. Casazza.
    The art of frame theory.
    Taiwanese J. Math., 4(2):129-201, 2000. MR 1757401 (2001f:42046)
  • 8. O. Christensen.
    An introduction to frames and Riesz bases.
    Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2003.MR 1946982 (2003k:42001)
  • 9. O. Christensen and T. Strohmer.
    The finite section method and problems in frame theory.
    Preprint, 2003.
  • 10. E. Cordero and K. Gröchenig.
    Localization of frames II.
    Appl. Comput. Harmon. Anal., 17:29-47, 2004. MR 2067914
  • 11. I. Daubechies.
    Ten lectures on wavelets.
    Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107 (93e:42045)
  • 12. S. Demko, W. F. Moss, and P. W. Smith.
    Decay rates for inverses of band matrices.
    Math. Comp., 43(168):491-499, 1984. MR 0758197 (85m:15002)
  • 13. H. G. Feichtinger.
    Gewichtsfunktionen auf lokalkompakten Gruppen.
    Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 188(8-10):451-471, 1979. MR 0599884 (81m:43004)
  • 14. H. G. Feichtinger and N. Kaiblinger.
    Perturbation of Gabor frames.
    Trans. Amer. Math. Soc., 356(5):2001-2023, 2004. MR 2031050 (2004k:42044)
  • 15. G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig, and C. Molitor-Braun.
    Weighted group algebras on groups of polynomial growth.
    Math. Z., 102(3):791-821, 2003. MR 2020712 (2004k:43008)
  • 16. I. Gel'fand, D. Raikov, and G. Shilov.
    Commutative normed rings.
    Chelsea Publishing Co., New York, 1964. MR 0205105 (34:4940)
  • 17. K. Gröchenig.
    Foundations of time-frequency analysis.
    Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717 (2002h:42001)
  • 18. K. Gröchenig.
    Localized frames are finite unions of Riesz sequences.
    Adv. Comput. Math., 18(2-4):149-157, 2003. MR 1968117 (2004a:42044)
  • 19. K. Gröchenig.
    Localization of frames, Banach frames, and the invertibility of the frame operator.
    J.Fourier Anal. Appl., 10(2):105-132, 2004. MR 2054304
  • 20. K. Gröchenig and M. Leinert.
    Wiener's lemma for twisted convolution and Gabor frames.
    J. Amer. Math. Soc., 17:1-18, 2004. MR 2015328 (2004m:42037)
  • 21. R. Hagen, S. Roch, and B. Silbermann.
    $ C\sp *$-algebras and numerical analysis, volume 236 of Monographs and Textbooks in Pure and Applied Mathematics.
    Marcel Dekker, Inc., New York, 2001. MR 1792428 (2002g:46133)
  • 22. C. E. Heil and D. F. Walnut.
    Continuous and discrete wavelet transforms.
    SIAM Rev., 31(4):628-666, 1989. MR 1025485 (91c:42032)
  • 23. A. Hulanicki.
    On the spectrum of convolution operators on groups with polynomial growth.
    Invent. Math., 17:135-142, 1972. MR 0323951 (48:2304)
  • 24. S. Jaffard.
    Propriétés des matrices ``bien localisées'' près de leur diagonale et quelques applications.
    Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(5):461-476, 1990.MR 1138533 (93h:47035)
  • 25. V. Losert.
    On the structure of groups with polynomial growth. II.
    J. London Math. Soc. (2), 63(3):640-654, 2001. MR 1825980 (2002f:22007)
  • 26. T. W. Palmer.
    Spectral algebras.
    Rocky Mountain J. Math., 22(1):293-328, 1992. MR 1159960 (93d:46079)
  • 27. T. W. Palmer.
    Banach algebras and the general theory of $ \sp *$-algebras. Vol. I, volume 49 of Encyclopedia of Mathematics and its Applications.
    Cambridge University Press, Cambridge, 1994. MR 1270014 (95c:46002)
  • 28. T. W. Palmer.
    Banach algebras and the general theory of $ *$-algebras. Vol. 2, volume 79 of Encyclopedia of Mathematics and its Applications.
    Cambridge University Press, Cambridge, 2001. MR 1819503 (2002e:46002)
  • 29. T. Pytlik.
    On the spectral radius of elements in group algebras.
    Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21:899-902, 1973.MR 0328476 (48:6818)
  • 30. M. Reed and B. Simon.
    Methods of modern mathematical physics. I.
    Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, second edition, 1980.MR 0751959 (85e:46002)
  • 31. W. Rudin.
    Functional analysis.
    McGraw-Hill Series in Higher Mathematics.
    McGraw-Hill Book Co., New York, 1973. MR 0365062 (51:1315)
  • 32. J. Sjöstrand.
    Wiener type algebras of pseudodifferential operators.
    Séminaire sur les Équations aux Dérivées Partielles, 1994-1995,
    Exp. No. IV, 21, École Polytech., Palaiseau, 1995. MR 1362552 (96j:47049)
  • 33. T. Strohmer.
    Rates of convergence for the approximation of shift-invariant systems in $ \ell ^2(\mathbb{Z})$.
    J. Fourier Anal. Appl., 5(6):599-616, 2000. MR 1752593 (2001b:42041)
  • 34. T. Strohmer.
    Approximation of dual Gabor frames, window decay, and wireless communications.
    Appl. Comput. Harmon. Anal., 11(2):243-262, 2001. MR 1848305 (2002j:42049)
  • 35. T. Strohmer.
    Four short stories about Toeplitz matrix calculations.
    Linear Algebra Appl., 343/344:321-344, 2002.
    Special issue on structured and infinite systems of linear equations. MR 1878948 (2002k:47060)

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Additional Information

Karlheinz Gröchenig
Affiliation: Department of Mathematics, The University of Connecticut, Storrs, Connecticut 06269-3009
Address at time of publication: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria

Michael Leinert
Affiliation: Fakultät für Mathematik, Institut für Angewandte Mathematik, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany

Keywords: Off-diagonal decay of matrices, weight function, invertibility of operators, invariance of spectrum, symmetric Banach algebras, Gelfand-Raikov-Shilov condition, Schur's test, twisted convolution, localized frames
Received by editor(s): November 4, 2003
Received by editor(s) in revised form: August 13, 2004
Published electronically: January 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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