The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators

Balan, Radu

doi:10.1090/S0002-9947-08-04448-6

The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators
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Trans. Amer. Math. Soc. 360 (2008), 3921-3941 Request permission

Abstract:

In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one $L^2$ function. We also estimate the coefficient decay of the inverse of finite linear combinations of time-frequency shifts.

References

Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau, Deficits and excesses of frames, Adv. Comput. Math. 18 (2003), no. 2-4, 93–116. Frames. MR 1968114, DOI 10.1023/A:1021360227672
Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau, Excesses of Gabor frames, Appl. Comput. Harmon. Anal. 14 (2003), no. 2, 87–106. MR 1981203, DOI 10.1016/S1063-5203(03)00006-X
Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau, Density, overcompleteness, and localization of frames. I. Theory, J. Fourier Anal. Appl. 12 (2006), no. 2, 105–143. MR 2224392, DOI 10.1007/s00041-006-6022-0
Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau, Density, overcompleteness, and localization of frames. II. Gabor systems, J. Fourier Anal. Appl. 12 (2006), no. 3, 309–344. MR 2235170, DOI 10.1007/s00041-005-5035-4
A. G. Baskakov, Estimates for the elements of inverse matrices, and the spectral analysis of linear operators, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 6, 3–26 (Russian, with Russian summary); English transl., Izv. Math. 61 (1997), no. 6, 1113–1135. MR 1609144, DOI 10.1070/im1997v061n06ABEH000164
Ingrid Daubechies, H. J. Landau, and Zeph Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl. 1 (1995), no. 4, 437–478. MR 1350701, DOI 10.1007/s00041-001-4018-3
Massimo Fornasier and Karlheinz Gröchenig, Intrinsic localization of frames, Constr. Approx. 22 (2005), no. 3, 395–415. MR 2164142, DOI 10.1007/s00365-004-0592-3
I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations Operator Theory 12 (1989), no. 3, 343–382. MR 998278, DOI 10.1007/BF01235737
K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston, Inc., 2001.
Karlheinz Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10 (2004), no. 2, 105–132. MR 2054304, DOI 10.1007/s00041-004-8007-1
Karlheinz Gröchenig, Time-frequency analysis of Sjöstrand’s class, Rev. Mat. Iberoam. 22 (2006), no. 2, 703–724. MR 2294795, DOI 10.4171/RMI/471
K. Gröchenig and M. Leinert, Wiener’s Lemma for Twisted Convolution and Gabor Frames, Journal of AMS 17 (2003), no. 1, 1–18.
Karlheinz Gröchenig and Michael Leinert, Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2695–2711. MR 2204052, DOI 10.1090/S0002-9947-06-03841-4
Christopher Heil, Linear independence of finite Gabor systems, Harmonic analysis and applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006, pp. 171–206. MR 2249310, DOI 10.1007/0-8176-4504-7_{9}
A. Hulanicki, On the spectral radius of hermitian elements in group algebras, Pacific J. Math. 18 (1966), 277–287. MR 198267
A. Hulanicki, On symmetry of group algebras of discrete nilpotent groups, Studia Math. 35 (1970), 207–219 (errata insert). MR 278082, DOI 10.4064/sm-35-2-207-219
A. Hulanicki, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972), 135–142. MR 323951, DOI 10.1007/BF01418936
A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), no. 4, 403–436. MR 1350700, DOI 10.1007/s00041-001-4017-4
Peter A. Linnell, von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3269–3277. MR 1637388, DOI 10.1090/S0002-9939-99-05102-3
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
Jean Ludwig, A class of symmetric and a class of Wiener group algebras, J. Functional Analysis 31 (1979), no. 2, 187–194. MR 525950, DOI 10.1016/0022-1236(79)90060-0
M. A. Naĭmark, Normed algebras, 3rd ed., Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the second Russian edition by Leo F. Boron. MR 0438123
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Marc A. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), no. 4, 403–418. MR 639575, DOI 10.1007/BF01465863
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. MR 1068530
Leiba Rodman, Ilya M. Spitkovsky, and Hugo J. Woerdeman, Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions, Trans. Amer. Math. Soc. 350 (1998), no. 6, 2185–2227. MR 1422908, DOI 10.1090/S0002-9947-98-01937-0
Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in $L_2(\mathbf R^d)$, Duke Math. J. 89 (1997), no. 2, 237–282. MR 1460623, DOI 10.1215/S0012-7094-97-08913-4
J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, École Polytech., Palaiseau, 1995, pp. Exp. No. IV, 21. MR 1362552
Thomas Strohmer, Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal. 20 (2006), no. 2, 237–249. MR 2207837, DOI 10.1016/j.acha.2005.06.003
—, personal communication (September 2005).
A. Zygmund, Trigonometric Series, University Press, Cambridge, 3rd Edition, 2003.