A hyperbolicity criterion for a class of diffeomorphisms of an infinite-dimensional torus

On an infinite-dimensional torus $\mathbb{T}^{\infty} = E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional real Banach space and $\mathbb{Z}^{\infty}$ is an abstract integer lattice, a special class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ is considered. It consists of the maps $G\colon \mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ equal to sums of invertible bounded linear operators preserving $\mathbb{Z}^{\infty}$ and $C^1$-smooth periodic additives. Necessary and sufficient conditions ensuring that such maps are hyperbolic (that is, are Anosov diffeomorphisms) are obtained.
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