Pseudo-Differential Operators on Z

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