Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

MULTIPLIERS FOR OPERATOR-VALUED BESSEL SEQUENCES AND GENERALIZED HILBERT-SCHMIDT CLASSES

  • KRISHNA, K. MAHESH (Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka(NITK)) ;
  • JOHNSON, P. SAM (Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka(NITK)) ;
  • MOHAPATRA, R.N. (Department of Mathematics, University of Central Florida)
  • Received : 2021.07.27
  • Accepted : 2021.09.19
  • Published : 2022.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

In 1960, Schatten studied operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(x_n{\otimes}{\bar{y_n}})$, where {xn}n and {yn}n are orthonormal sequences in a Hilbert space, and {λn}n ∈ ℓ∞(ℕ). Balazs generalized some of the results of Schatten in 2007. In this paper, we further generalize results of Balazs by studying the operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(A^*_nx_n{\otimes}{\bar{B^*_ny_n}})$, where {An}n and {Bn}n are operator-valued Bessel sequences, {xn}n and {yn}n are sequences in the Hilbert space such that {║xn║║yn║}n ∈ ℓ∞(ℕ). We also generalize the class of Hilbert-Schmidt operators studied by Balazs.

Keywords

Acknowledgement

The first author thanks the National Institute of Technology Karnataka (NITK), Surathkal for giving financial support. The third author is grateful to the Mohapatra Family Foundation for their support to pursue this research.

References

  1. Albrecht Pietsch, Operator ideals, North-Holland Publishing Co., Amsterdam-New York, 1980.
  2. Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1993.
  3. David L. Donoho and Michael Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization, Proc. Natl. Acad. Sci. USA 100 (2003), 2197-2202. https://doi.org/10.1073/pnas.0437847100
  4. Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), x+94.
  5. Diana T. Stoeva and Peter Balazs, Detailed characterization of conditions for the unconditional convergence and invertibility of multipliers, Sampl. Theory Signal Image Process. 12 (2013), 87-125. https://doi.org/10.1007/BF03549563
  6. Diana T. Stoeva and Peter Balazs, Riesz bases multipliers, Oper. Theory Adv. Appl. 236 (2014), 475-482.
  7. Diana T. Stoeva and Peter Balazs, On the dual frame induced by an invertible frame multiplier, Sampl. Theory Signal Image Process. 15 (2016), 119-130. https://doi.org/10.1007/BF03549600
  8. R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366. https://doi.org/10.1090/S0002-9947-1952-0047179-6
  9. D. Gabor, Theory of Communication, J. IEE 93 (1946), 429-457.
  10. Ingrid Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283. https://doi.org/10.1063/1.527388
  11. Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.
  12. John R. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold Co., London, 1971.
  13. Karlheinz Grochenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhauser Boston, Inc., Boston, MA, 2001.
  14. Karlheinz Grochenig and Christopher Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory 34 (1999), 439-457. https://doi.org/10.1007/BF01272884
  15. Michael Frank, Vern I. Paulsen, and Terry R. Tiballi, Symmetric approximation of frames and bases in Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002), 777-793. https://doi.org/10.1090/S0002-9947-01-02838-0
  16. Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, [Cham], second edition, 2016.
  17. Peter Balazs and D.T. Stoeva, Representation of the inverse of a frame multiplier, J. Math. Anal. Appl. 442 (2015), 981-994.
  18. Peter Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (2007), 571-585. https://doi.org/10.1016/j.jmaa.2006.02.012
  19. Peter Balazs, Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), 315-330. https://doi.org/10.1142/S0219691308002379
  20. Robert W. Heath Jr. and Arogyaswami Paulraj, Linear dispersion codes for MIMO systems based on frame theory, IEEE Transactions on Signal Processing 50 (2002), 2429-2441. https://doi.org/10.1109/TSP.2002.803325
  21. Robert Schatten, On reflexive norms for the direct product of Banach spaces, Trans. Amer. Math. Soc. 54 (1943), 498-506. https://doi.org/10.1090/S0002-9947-1943-0009086-9
  22. Robert Schatten, On the direct product of Banach spaces, Trans. Amer. Math. Soc. 53 (1943), 195-217. https://doi.org/10.1090/S0002-9947-1943-0007568-7
  23. Robert Schatten, The cross-space of linear transformations, Ann. of Math. 2 (1946), 73-84. https://doi.org/10.2307/1969036
  24. Robert Schatten, Norm ideals of completely continuous operators, Second printing. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 27, Springer-Verlag, Berlin-New York, 1970.
  25. Robert Schatten and John von Neumann, The cross-space of linear transformations, Ann. of Math. 2 (1946), 608-630.
  26. Robert Schatten and John von Neumann, The cross-space of linear transformations, Ann. of Math. 2 (1948), 557-582.
  27. D.T. Stoeva and P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal. 33 (2012), 292-299. https://doi.org/10.1016/j.acha.2011.11.001
  28. Wenchang Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437-452. https://doi.org/10.1016/j.jmaa.2005.09.039
  29. Yonina C. Eldar and G. David Forney, Optimal tight frames and quantum measurement, IEEE Trans. Inform. Theory 48 (2002), 599-610. https://doi.org/10.1109/18.985949