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Pseudo-Differential Operators on ℤ

  • Shahla Molahajloo
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 205)

Abstract

A necessary and sufficient condition is imposed on the symbols σ: ℤ×S 1→ℂ to guarantee that the corresponding pseudo-differential operators T σ: L 2(ℤ)→L 2(ℤ) are Hilbert-Schmidt. A special sufficient condition on the symbols σ: ℤ×S 1→ℂ for the corresponding pseudo-differential operators T σ: L 2(ℤ)→L 2(ℤ) to be bounded is given. Sufficient conditions are given on the symbols σ: ℤ×S 1→ℂ to ensure the boundedness and compactness of the corresponding pseudo-differential operators T σ: L p (ℤ)→L p (ℤ) for 1≨p<∞. Norm estimates for the pseudo-differential operators T σ are given in terms of the symbols σ. The almost diagonalization of the pseudo-differential operators is then shown to follow from the sufficient condition for the L p -boundedness.

Keywords

Pseudo-differential operators Hilbert-Schmidt operators L2-boundedness Lp-boundedness Lp-compactness almost diagonalization 

Mathematics Subject Classification (2000)

Primary 47G30 Secondary 65R10 65T50 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Shahla Molahajloo
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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