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Remarks on Naimark's duality

Author(s): Wojciech Czaja
Journal: Proc. Amer. Math. Soc. 136 (2008), 867-871.
MSC (2000): Primary 42C15
Posted: November 30, 2007
MathSciNet review: 2361858
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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:

1.: N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications, Inc., New York, NY, 1993. MR 1255973 (94i:47001)
2.: A. Aldroubi, Portraits of frames, Proc. Amer. Math. Soc. 123 (1993), no. 6, pp. 1661-1668. MR 1242070 (95g:46037)
3.: R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Deficits and excesses of frames. Frames, Adv. Comput. Math. 18 (2003), no. 2-4, pp. 93-116. MR 1968114 (2004a:42040)
4.: R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Excesses of Gabor frames, Appl. Comput. Harmon. Anal. 14 (2003), no. 2, pp. 87-106. MR 1981203 (2004c:42058)
5.: N. K. Bari, Sur les bases dans l'espace de Hilbert, Dokl. Akad. Nauk. SSSR 54 (1946), pp. 379-382. MR 0020169 (8:513a)
6.: N. K. Bari, Biorthogonal systems and bases in Hilbert spaces, Uchen. Zap. Moskov. Gos. Univ. 148 (1951), pp. 69-107. MR 0050171 (14:289b)
7.: J. J. Benedetto, C. Heil, and D. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl. 1 (1995), no. 4, pp. 355-402. MR 1350699 (96f:42002)
8.: P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in The functional and harmonic analysis of wavelets and frames, pp. 149-182, Contemp. Math., 247, Amer. Math. Soc., Providence, RI, 1999. MR 1738089 (2000m:46015)
9.: P. G. Casazza and J.Kovačević, Equal-norm tight frames with erasures. Frames, Adv. Comput. Math. 18 (2003), no. 2-4, pp. 387-430. MR 1968127 (2004e:42046)
10.: P. G. Casazza, G. Kutyniok, and M. C. Lammers, Duality principles in frame theory, J. Fourier Anal. Appl. 10 (2004), pp. 383-408. MR 2078264 (2005d:42033)
11.: C. Foias and B. Sz.-Nagy, Harmonic analysis of operators on Hilbert space, American Elsevier Publishing Co., Inc., New York, NY, 1970. MR 0275190 (43:947)
12.: D. Han and D. R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697. MR 1686653 (2001a:47013)
13.: C. Houdré, Wavelets, probability, and statistics: some bridges, in Wavelets: mathematics and applications, pp. 365-398, Stud. Adv. Math., CRC, Boca Raton, FL, 1994. MR 1247521 (95c:60046)
14.: B. S. Kashin and T. Yu. Kulikova, A note on the description of frames of general form, Mathematical Notes, vol. 72 (2002), no. 6, pp. 863-867. MR 1964152 (2003m:46034)
15.: D. R. Larson, Unitary systems, wavelet sets, and operator-theoretic interpolation of wavelets and frames, arXiv Math.FA/0604615
16.: D. R. Larson, Unitary systems and wavelet sets, in Wavelet Analysis and Applications, Birkhäuser, Basel (2007), pp. 143-171. MR 2297918
17.: M. A. Naimark, Spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat., vol. 4 (1940), no. 3, pp. 277-318. MR 0002714 (2:105a)
18.: M. A. Naimark, On a representation of additive operator set functions, Dokl. Acad. Sci. SSSR, vol. 41 (1943), no. 9, pp. 373-375.
19.: A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in $L_2(\R^d)$ , Duke Math. J. 89 (1997), no. 2, pp. 237-282. MR 1460623 (98i:42013)
20.: P. A. Terekhin, Representation systems and projections of bases, Mathematical Notes, vol. 75 (2004), no. 6, pp. 881-884. MR 2086620 (2005e:46014)

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Additional Information:

Wojciech Czaja
Affiliation: Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: wojtek@math.umd.edu

DOI: 10.1090/S0002-9939-07-09048-X
PII: S 0002-9939(07)09048-X
Keywords: Naimark dilation theorem, frame, Bessel system, complete system, Riesz basis, representation system, Schauder basis, linearly independent system
Received by editor(s): January 3, 2005
Received by editor(s) in revised form: April 26, 2006
Posted: November 30, 2007
Additional Notes: The author was supported by Marie Curie Intra-European Fellowship FP6-2003-500685
Communicated by: David R. Larson
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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