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Uniform exponential (UE) stability of linear difference equations withinfinite delay is studied using the notions of a stability radius and a phasespace. The state space $\X$ is supposed to be an abstract Banach space. We workboth 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 areobtained 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 orcompactness assumptions on coefficients $K(j)$. Time-varying (non-convolution)difference equations are studied via structured UE stability radii $\r_\t$ ofconvolution equations. These radii correspond to a feedback scheme with delayedoutput and time-varying disturbances. We also consider stability radii $\r_\c$associated with a time-invariant disturbance operator, unstructured stabilityradii, and stability radii corresponding to delayed feedback. For all thesetypes of stability radii two-sided estimates are obtained. The estimates fromabove are given in terms of the Z-transform $\wh K (\zeta)$, the estimate frombelow via the norm of the input-output operator. These estimates turn intoexplicit formulae if the state space $\X$ is Hilbert or if disturbances aretime-invariant. The results on stability radii are applied to obtain variousexponential stability tests for non-convolution equations. Several examples areprovided.

Structured stability radii and exponential stability tests for Volterra difference systems