Advances in Operator Theory

Characterization of $K$-frame vectors and $K$-frame generator multipliers

Somayeh Javani and Farkhondeh Takhteh

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


‎‎Let $\mathcal{U}$ be a unitary system and let $\mathcal{B(U)}$ be the Bessel vector space for $\mathcal{U}$‎. ‎In this article‎, ‎we give a characterization of Bessel vector spaces and local commutant spaces at different complete frame vectors‎. ‎The relation between local commutant spaces at different complete frame vectors is investigated‎. ‎Moreover‎, ‎by introducing multiplication and adjoint on the Bessel vector space for a unital semigroup of unitary operators‎, ‎we give a $C^*$-algebra structure to $\mathcal{B(U)}$‎. ‎Then‎, ‎we construct some subsets of $K$-frame vectors that have a Banach space or Banach algebra structure‎. ‎Also‎, ‎as a consequence‎, ‎the set of complete frame vectors for different unitary systems contains Banach spaces or Banach algebras‎. ‎In the end‎, ‎we give several characterizations of $K$-frame generator multipliers and Parseval $K$-frame generator multipliers‎.

Article information

Adv. Oper. Theory, Volume 4, Number 3 (2019), 587-603.

Received: 15 August 2018
Accepted: 18 December 2018
First available in Project Euclid: 2 March 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C15: General harmonic expansions, frames
Secondary: 42A38‎ ‎41A58

Bessel vector‎ ‎‎complete frame vector ‎‎unitary system $K$-frame vector‎ Bessel generator multiplier


Javani, Somayeh; Takhteh, Farkhondeh. Characterization of $K$-frame vectors and $K$-frame generator multipliers. Adv. Oper. Theory 4 (2019), no. 3, 587--603. doi:10.15352/aot.1808-1408.

Export citation


  • H. Bolcskei, F. Hlawatsch, and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process. 46 (1998), no. 12, 3256–3268.
  • W. Consortium, Basic properties of wavelets, J. Fourier Anal. Appl. 4 (1998), no. 4-5, 575–594.
  • X. Dai and D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no. 640, viii+68 pp.
  • R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), no. 2, 413–415.
  • R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), no. 2, 341–366.
  • Y. C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier Anal. Appl. 9 (2003), no. 1, 77–96.
  • P. A. Fillmore and J. P. Williams, On operator ranges, Adv. Math. 7 (1971), no. 3, 254–281.
  • L. Găvruţa, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012), no. 1, 139–144.
  • X. Guo, Canonical dual $K$-Bessel sequences and dual $K$-Bessel generators for unitary systems of Hilbert spaces, J. Math. Anal. Appl. 444 (2016), no. 1, 598–609.
  • D. Han, Wandering vectors for irrational rotation unitary systems, Trans. Amer. Math. Soc., 350 (1998), no.1, 309–320.
  • D. Han and D. R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94 pp.
  • D. Han and D. R. Larson, Wandering vector multipliers for unitary groups, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3347–3370.
  • D. Han and D. R. Larson, Unitary systems and Bessel generator multipliers, Wavelets and multiscale analysis, 131–150, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2011.
  • G. Ji and K. S. Saito, On wandering vector multipliers for unitary groups, Proc. Amer. Math. Soc., 133 (2005), no. 11, 3263–3269.
  • D. R. Larson, Unitary systems and wavelet sets, Wavelet analysis and applications, 143–171, Appl. Numer. Harmon. Anal., Birkhäuser, Basel, 2007.
  • Z. Li, X. Dai, Y. Diao, and J. Xin, Multipliers, phases and connectivity of mra wavelets in $L^2(\Bbb R^2)$, J. Fourier Anal. Appl., 16 (2010), no. 2, 155–176.
  • G. J. Murphy, $C^*$-algebras and operator theory, Academic press, 2014.
  • W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg,1973.
  • Z. Q. Xiang and Y. M. Li, Frame sequences and dual frames for operators, Science Asia 42 (2016), no. 3, 222–230.
  • X. Xiao, Y. Zhu, and L. Găvruţa, Some properties of $K$-frames in Hilbert spaces, Results Math. 63 (2013), no. 3-4, 1243–1255.