Skip to main content
Log in

Frames and weak frames for unbounded operators

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In 2012, Găvruţa introduced the notions of K-frame and of atomic system for a linear bounded operator K in a Hilbert space \({\mathcal{H}}\), in order to decompose its range \(\mathcal {R}(K)\) with a frame-like expansion. In this article, we revisit these concepts for an unbounded and densely defined operator \(A:\mathcal {D}(A)\to {\mathcal{H}}\) in two different ways. In one case, we consider a non-Bessel sequence where the coefficient sequence depends continuously on \(f\in \mathcal {D}(A)\) with respect to the norm of \({\mathcal{H}}\). In the other case, we consider a Bessel sequence and the coefficient sequence depends continuously on \(f\in \mathcal {D}(A)\) with respect to the graph norm of A.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Antoine, J.-P., Balazs, P.: Frames and semi-frames. J. Phys. A: Math. Theor. 44, 205201 (2011). Corrigendum 44,(2011) 479501

    Article  MathSciNet  Google Scholar 

  2. Antoine, J.-P., Balazs, P.: Frames, semi-frames, and Hilbert scales. Numer. Funct. Anal. Optim. 33, 736–769 (2012)

    Article  MathSciNet  Google Scholar 

  3. Antoine, J.-P., Balazs, P., Stoeva, D.T.: Classification of general sequences by frame related operators. Sampling Theory Signal Image Proc. 10, 151–170 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Antoine, J.-P., Inoue, A., Trapani, C.: Partial *-Algebras and their Operator Realizations. Kluwer, Dordrecht (2002)

    Book  Google Scholar 

  5. Antoine, J.-P., Speckbacher, M., Trapani, C.: Reproducing pairs of measurable functions. Acta Appl. Math. 150, 81–101 (2017)

    Article  MathSciNet  Google Scholar 

  6. Antoine, J.-P., Trapani, C.: Reproducing pairs of measurable functions and partial inner product spaces. Adv. Operator Th. 2, 126–146 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Bagarello, F., Bellomonte, G.: Hamiltonians defined by biorthogonal sets. J. Phys. A: Math. Theor. 50(14), 145203 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bagarello, F., Inoue, A., Trapani, C.: Non-self-adjoint hamiltonians defined by Riesz bases. J. Math. Phys. 55, 033501 (2014)

    Article  MathSciNet  Google Scholar 

  9. Bagarello, F., Inoue, H., Trapani, C.: Biorthogonal vectors, sesquilinear forms, and some physical operators. J. Math. Phys. 59, 033506 (2018)

    Article  MathSciNet  Google Scholar 

  10. Balazs, P., Speckbacher, M.: Reproducing pairs and Gabor systems at critical density. J. Math. Anal Appl. 455(2), 1072–1087 (2017)

    Article  MathSciNet  Google Scholar 

  11. Balazs, P., Speckbacher, M.: Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups. J. Phys. A: Math. Theor. 48, 395201 (2015)

    Article  MathSciNet  Google Scholar 

  12. Bellomonte, G., Trapani, C.: Riesz-like bases in rigged Hilbert spaces. Z. Anal. Anwend. 35, 243–265 (2016)

    Article  MathSciNet  Google Scholar 

  13. Beutler, F.J., Root, W.L.: The operator pseudoinverse in control and systems identification. In: Zuhair Nashed, M. (ed.) Generalized Inverses and Applications. Academic Press, New York (1976)

  14. Casazza, P., Christensen, O., Li, S., Lindner, A.: Riesz-Fischer sequences and lower frame bounds. Z. Anal. Anwend. 21(2), 305–314 (2002)

    Article  MathSciNet  Google Scholar 

  15. Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl. 195, 401–414 (1995)

    Article  MathSciNet  Google Scholar 

  16. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)

    Book  Google Scholar 

  17. Corso, R.: Sesquilinear forms associated to sequences on Hilbert spaces. Monatsh Math. 189(4), 625–650 (2019)

    Article  MathSciNet  Google Scholar 

  18. Corso, R.: Generalized frame operator, lower semi-frames and sequences of translates. arXiv:1912.03261 (2019)

  19. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)

    Article  MathSciNet  Google Scholar 

  20. Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17(1), 413–415 (1966)

    Article  MathSciNet  Google Scholar 

  21. Duffin, J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  Google Scholar 

  22. Feichtinger, H.G., Werther, T.: Atomic systems for subspaces. In: Zayed, L. (ed.) Proceedings SampTA 2001, Orlando, FL, pp 163–165 (2001)

  23. Găvruţa, L.: Frames for operators. Appl. Comp. Harmon. Anal. 32, 139–144 (2012)

    Article  MathSciNet  Google Scholar 

  24. Geddavalasa, R., Johnson, P.S.: Frames for operators in Banach spaces. Acta Math. Vietnam. 42(4), 665–673 (2017)

    Article  MathSciNet  Google Scholar 

  25. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  26. Guo, X.: Canonical dual K-Bessel sequences and dual K-Bessel generators for unitary systems of Hilbert spaces. J. Math. Anal Appl. 444, 598–609 (2016)

    Article  MathSciNet  Google Scholar 

  27. Heil, C.: A Basis Theory Primer: Expanded Edition. Birkhäuser, Boston (2011)

    Book  Google Scholar 

  28. Inoue, H., Takakura, M.: Non-self-adjoint Hamiltonians defined by generalized Riesz bases. J. Math. Phys. 57, 083505 (2016)

    Article  MathSciNet  Google Scholar 

  29. Javanshiri, H., Fattahi, A.-M.: Continuous atomic systems for subspaces. Mediterr. J. Math. 13(4), 1871–1884 (2016)

    Article  MathSciNet  Google Scholar 

  30. Li, S., Ogawa, H.: Pseudo-duals of frames with applications. Appl. Comput. Harm. Anal. 11, 289–304 (2001)

    Article  MathSciNet  Google Scholar 

  31. Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10(4), 409–431 (2004)

    Article  MathSciNet  Google Scholar 

  32. Najati, A., Mohammadi Saem, M., Găvruţa, P.: Frames and operators in Hilbert C*-modules. Operators and Matrices 10(1), 73–81 (2016)

    Article  MathSciNet  Google Scholar 

  33. Neyshaburi, F.A., Arefijamaal, A.A.: Some constructions of K-frames and their duals. Rocky Mountain J. Math. 47(6), 1749–1764 (2017)

    Article  MathSciNet  Google Scholar 

  34. Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Springer, Dordrecht (2012)

    Book  Google Scholar 

  35. Stoeva, D.T., Balazs, P.: A survey on the unconditional convergence and the invertibility of multipliers with implementation. In: S. D. Casey, K. Okoudjou, M. Robinson, B. Sadler (Eds.) Sampling - Theory and Applications (A Centennial Celebration of Claude Shannon), Applied and Numerical Harmonic Analysis Series, Springer, (accepted) (2018)

  36. Xiao, X., Zhu, Y., Găvruţa, L.: Some properties of K-frames in Hilbert spaces. Results Math. 63, 1243–1255 (2013)

    Article  MathSciNet  Google Scholar 

  37. Young, R.: An Introduction to Nonharmonic Fourier Series. Academic, New York (1980). (revised first edition 2001)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors warmly thank Prof. C. Trapani and the referees for their fruitful comments and remarks.

Funding

This work has been in part financially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Giorgia Bellomonte.

Additional information

Communicated by: Tomas Sauer

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bellomonte, G., Corso, R. Frames and weak frames for unbounded operators. Adv Comput Math 46, 38 (2020). https://doi.org/10.1007/s10444-020-09773-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-020-09773-3

Keywords

Mathematics Subject Classification (2010)

Navigation