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

Published: 29 March 2022

Abstract: The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on $\mathbb {Z}^n\times \mathbb {T}^n$ 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 $\mathbb {Z}^n$. The index of an elliptic pseudo-differential operator on $\mathbb {Z}^n$ is also computed.
Keywords: Pseudo-differential operators / minimal and maximal operators / ellipticity / Fredholmness / index

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