Asymptotics of solutions of one-dimensional difference equations with constant operator coefficients

The authors study equations of the form
$$ \sum_{k\geqslant0}A_ku_{n-k}=f_n,\qquad n=0,\pm1,\dots, $$
where the $u_n$ and $f_n$ are elements in some Hilbert space $H$, and the $A_k$ are bounded linear operators on $H$. It is assumed that the corresponding operator symbol
$$ L(\lambda )=\sum_{k\geqslant0}A_k\lambda^k $$
is a holomorphic Fredholm operator-valued function which is normal in some neighborhood of zero.
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