Remarks on Naimark’s duality
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Abstract:
We present an extension of a version of Naimark’s dilation theorem which states that complete systems in a Hilbert space are projections of $\omega$-linearly independent systems of elements of an ambient Hilbert space. This result is presented in the context of other known extensions of Naimark’s theorem.References
- N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications, Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell; Reprint of the 1961 and 1963 translations; Two volumes bound as one. MR 1255973
- Akram Aldroubi, Portraits of frames, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1661–1668. MR 1242070, DOI 10.1090/S0002-9939-1995-1242070-5
- 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
- N. Bary, Sur les bases dans l’espace de Hilbert, C. R. (Doklady) Acad. Sci. URSS (N. S.) 54 (1946), 379–382 (French). MR 0020169
- N. K. Bari, Biorthogonal systems and bases in Hilbert space, Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4) (1951), 69–107 (Russian). MR 0050171
- John J. Benedetto, Christopher Heil, and David F. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl. 1 (1995), no. 4, 355–402. MR 1350699, DOI 10.1007/s00041-001-4016-5
- Peter G. Casazza, Deguang Han, and David R. Larson, Frames for Banach spaces, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 149–182. MR 1738089, DOI 10.1090/conm/247/03801
- Peter G. Casazza and Jelena Kovačević, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003), no. 2-4, 387–430. Frames. MR 1968127, DOI 10.1023/A:1021349819855
- Peter G. Casazza, Gitta Kutyniok, and Mark C. Lammers, Duality principles in frame theory, J. Fourier Anal. Appl. 10 (2004), no. 4, 383–408. MR 2078264, DOI 10.1007/s00041-004-3024-7
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653, DOI 10.1090/memo/0697
- Christian Houdré, Wavelets, probability, and statistics: some bridges, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 365–398. MR 1247521
- B. S. Kashin and T. Yu. Kulikova, A remark on the description of frames of general form, Mat. Zametki 72 (2002), no. 6, 941–945 (Russian); English transl., Math. Notes 72 (2002), no. 5-6, 863–867. MR 1964152, DOI 10.1023/A:1021402315905
- D. R. Larson, Unitary systems, wavelet sets, and operator-theoretic interpolation of wavelets and frames, arXiv Math.FA/0604615
- David R. Larson, Unitary systems and wavelet sets, Wavelet analysis and applications, Appl. Numer. Harmon. Anal., Birkhäuser, Basel, 2007, pp. 143–171. MR 2297918, DOI 10.1007/978-3-7643-7778-6_{1}4
- M. Neumark, Spectral functions of a symmetric operator, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 277–318 (Russian, with English summary). MR 0002714
- M. A. Naimark, On a representation of additive operator set functions, Dokl. Acad. Sci. SSSR, vol. 41 (1943), no. 9, pp. 373–375.
- 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
- P. A. Terëkhin, Representation systems and projections of bases, Mat. Zametki 75 (2004), no. 6, 944–947 (Russian); English transl., Math. Notes 75 (2004), no. 5-6, 881–884. MR 2086620, DOI 10.1023/B:MATN.0000030997.11701.c0
Additional Information
- Wojciech Czaja
- Affiliation: Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
- Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Email: wojtek@math.umd.edu
- Received by editor(s): January 3, 2005
- Received by editor(s) in revised form: April 26, 2006
- Published electronically: November 30, 2007
- Additional Notes: The author was supported by Marie Curie Intra-European Fellowship FP6-2003-500685
- Communicated by: David R. Larson
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 867-871
- MSC (2000): Primary 42C15
- DOI: https://doi.org/10.1090/S0002-9939-07-09048-X
- MathSciNet review: 2361858