Pseudo-Differential Operators on ℤ.

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: L²(ℤ)→L²(ℤ) are Hilbert-Schmidt. A special sufficient condition on the symbols σ: ℤ×S ¹→ℂ for the corresponding pseudo-differential operators T_σ: L²(ℤ)→L²(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. [ABSTRACT FROM AUTHOR]