Taiwanese Journal of Mathematics

SHIFT PRESERVING OPERATORS ON LOCALLY COMPACT ABELIAN GROUPS

R. A. Kamyabi Gol and R. Raisi Tousi

Full-text: Open access

Abstract

We investigate shift preserving operators on locally compact abelian groups. We show that there is a one-to-one correspondence between shift preserving operators and range operators on $L^2(G)$ where $G$ is a locally compact abelian group. We conclude that a shift preserving operator has several properties in common with its associated range operator, especially compactness of one implies compactness of the other. Moreover, we obtain a necessary condition for a shift preserving operator to be Hilbert Schmidt or of finite trace in terms of its range function.

Article information

Source
Taiwanese J. Math. Volume 15, Number 5 (2011), 1939-1955.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406415

Digital Object Identifier
doi:10.11650/twjm/1500406415

Zentralblatt MATH identifier
1275.47018

Subjects
Primary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Secondary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Keywords
shift invariant space range function shift preserving operator range operator locally compact abelian group compact operator Hilbert Schmidt operator trace

Citation

Kamyabi Gol, R. A.; Raisi Tousi, R. SHIFT PRESERVING OPERATORS ON LOCALLY COMPACT ABELIAN GROUPS. Taiwanese J. Math. 15 (2011), no. 5, 1939--1955. doi:10.11650/twjm/1500406415. https://projecteuclid.org/euclid.twjm/1500406415


Export citation

References

  • A. Aldroubi, Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces, Appl. Comput. Harmon. Anal., 13 (2002), 151-161.
  • C. de Boor, R. A. DeVore and A. Ron, The structure of finitely generated shift invariant spaces in $L^2(\mathbb R^d)$, J. Funct. Anal., 119 (1994), 37-78.
  • M. Bownik, The structure of shift invariant subspaces of $L^2(\mathbb R^n)$, J. Funct. Anal., 177(2) (2000), 282-309.
  • C. S. Burrus, R. A. Gopinath and H. Guo, Introduction to Wavelets and Wavelet Transforms, Princetone Hall, Inc. USA, CRC Press, 1995.
  • L. Debnath, Wavelet Transforms and Their Applications, Birkhaüser, 2001.
  • G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
  • H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transform, Springer Lecture Notes in Mathematics, Nr. 1863, Berlin, 2005.
  • H. Helson, Lectures on Invariant Subspaces, Academic Press, New York, London, 1964.
  • E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer-Verlag, 1963.
  • D. Hong, J. Wang and R. Gardner, Real Analysis with an Introduction to Wavelets and Applications, Elsevier Academic Press, USA, 2005.
  • R. A. Kamyabi Gol and R. Raisi Tousi, A range function approach to shift-invariant spaces on locally compact abelian groups, Int. J. Wavelets. Multiresolut. Inf. Process, 8 (2010), 49-59.
  • R. A. Kamyabi Gol and R. Raisi Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340 (2008), 219-225
  • G. Kutyniok, Time Frequency Analysis on Locally Compact Groups, Ph.D. thesis, Padeborn University, (2000).
  • G. J. Murphy, $C^*$-Algebras and Operator Theory, Academic Press, Boston, 1990.
  • A. Ron and Z. Shen, Affine systems in $L^2(\mathbb{R}^d)$, the analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408-447.
  • A. Ron and Z. Shen, Frames and stable bases for shift invariant subspaces of $L^2(\mathbb R^d)$, Canad. J. Math., 47 (1995), 1051-1094.
  • A. Ron and Z. Shen, Generalized shift invariant systems, Constructive Approximation, 22 (2005), 1-45.
  • Z. Rzeszotnik, Characterization theorems in the theory of wavelets, Ph.D. thesis, Washingtone University, 2000.
  • R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, Berlin, 1970.