  Uniform exponential (UE) stability of linear difference equations with
infinite delay is studied using the notions of a stability radius and a phase
space. The state space $\X$ is supposed to be an abstract Banach space. We work
both with non-fading phase spaces $c_0 (\ZZ^-, \X)$ and $\ll^\infty (\ZZ^-,
\X)$ and with exponentially fading phase spaces of the $\ll^p$ and $c_0$ types.
For equations of the convolution type, several criteria of UE stability are
obtained in terms of the Z-transform $\wh K (\zeta)$ of the convolution kernel
$K (\cdot)$, in terms of the input-state operator and of the resolvent
(fundamental) matrix. These criteria do not impose additional positivity or
compactness assumptions on coefficients $K(j)$. Time-varying (non-convolution)
difference equations are studied via structured UE stability radii $\r_\t$ of
convolution equations. These radii correspond to a feedback scheme with delayed
output and time-varying disturbances. We also consider stability radii $\r_\c$
associated with a time-invariant disturbance operator, unstructured stability
radii, and stability radii corresponding to delayed feedback. For all these
types of stability radii two-sided estimates are obtained. The estimates from
above are given in terms of the Z-transform $\wh K (\zeta)$, the estimate from
below via the norm of the input-output operator. These estimates turn into
explicit formulae if the state space $\X$ is Hilbert or if disturbances are
time-invariant. The results on stability radii are applied to obtain various
exponential stability tests for non-convolution equations. Several examples are
provided.
