Lyapunov’s theorem for continuous frames

Bownik, Marcin

doi:10.1090/proc/14088

Lyapunov’s theorem for continuous frames
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by Marcin Bownik PDF

Proc. Amer. Math. Soc. 146 (2018), 3825-3838 Request permission

Abstract:

Akemann and Weaver (2014) have shown a remarkable extension of Weaver’s $KS_r$ Conjecture (2004) in the form of approximate Lyapunov’s theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov’s theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem.

References

Charles A. Akemann and Joel Anderson, Lyapunov theorems for operator algebras, Mem. Amer. Math. Soc. 94 (1991), no. 458, iv+88. MR 1086563, DOI 10.1090/memo/0458
Charles Akemann and Nik Weaver, A Lyapunov-type theorem from Kadison-Singer, Bull. Lond. Math. Soc. 46 (2014), no. 3, 517–524. MR 3210706, DOI 10.1112/blms/bdu005
S. Twareque Ali, J.-P. Antoine, and J.-P. Gazeau, Continuous frames in Hilbert space, Ann. Physics 222 (1993), no. 1, 1–37. MR 1206084, DOI 10.1006/aphy.1993.1016
Syed Twareque Ali, Jean-Pierre Antoine, and Jean-Pierre Gazeau, Coherent states, wavelets, and their generalizations, 2nd ed., Theoretical and Mathematical Physics, Springer, New York, 2014. MR 3154614, DOI 10.1007/978-1-4614-8535-3
Peter Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (2007), no. 1, 571–585. MR 2273547, DOI 10.1016/j.jmaa.2006.02.012
P. Balazs, D. Bayer, and A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Phys. A 45 (2012), no. 24, 244023, 20. MR 2930518, DOI 10.1088/1751-8113/45/24/244023
V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
M. Bownik, The Kadison-Singer problem, Frames and Harmonic Analysis, Contemp. Math. (to appear).
M. Bownik, P. Casazza, A. Marcus, and D. Speegle, Improved bounds in Weaver and Feichtinger conjectures, J. Reine Angew. Math. (to appear).
Peter G. Casazza and Janet Crandell Tremain, The Kadison-Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (2006), no. 7, 2032–2039. MR 2204073, DOI 10.1073/pnas.0507888103
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
Massimo Fornasier and Holger Rauhut, Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl. 11 (2005), no. 3, 245–287. MR 2167169, DOI 10.1007/s00041-005-4053-6
D. Freeman and D. Speegle, The discretization problem for continuous frames, preprint (2016) available at arXiv:1611.06469.
Jean-Pierre Gabardo and Deguang Han, Frames associated with measurable spaces, Adv. Comput. Math. 18 (2003), no. 2-4, 127–147. Frames. MR 1968116, DOI 10.1023/A:1021312429186
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
V. M. Kadets and G. Shekhtman, Lyapunov’s theorem for $l_p$-valued measures, Algebra i Analiz 4 (1992), no. 5, 148–154 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 5, 961–966. MR 1202728
Richard V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383–400. MR 123922, DOI 10.2307/2372748
Gerald Kaiser, A friendly guide to wavelets, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1287849
Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015), no. 1, 327–350. MR 3374963, DOI 10.4007/annals.2015.182.1.8
Bill Moran, Stephen Howard, and Doug Cochran, Positive-operator-valued measures: a general setting for frames, Excursions in harmonic analysis. Volume 2, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2013, pp. 49–64. MR 3050314, DOI 10.1007/978-0-8176-8379-5_{4}
J. J. Uhl Jr., The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969), 158–163. MR 264029, DOI 10.1090/S0002-9939-1969-0264029-1
Nik Weaver, The Kadison-Singer problem in discrepancy theory, Discrete Math. 278 (2004), no. 1-3, 227–239. MR 2035401, DOI 10.1016/S0012-365X(03)00253-X