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Ellipticity and Fredholmness of pseudo-differential operators on ℓ²(ℤⁿ)

Abstract: The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on Zn×Tn\mathbb {Z}^n\times \mathbb {T}^n are proved to coincide and the domain is given in terms of a Sobolev space. Ellipticity and Fredholmness are proved to be equivalent for pseudo-differential operators on Zn\mathbb {Z}^n. The index of an elliptic pseudo-differential operator on Zn\mathbb {Z}^n is also computed.
Keywords: Pseudo-differential operators / minimal and maximal operators / ellipticity / Fredholmness / index
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