The Volterra difference equation on a Banach space \(X,\) \[ x\left( n+1\right) =\sum_{j=-\infty }^{n}Q\left( n-j\right) x\left( j\right) ,\quad n\in \mathbb{Z}^{+}=\{0,1,2,\dots\} \] is treated, where \(Q(n) \) are bounded linear operators on \(X\) such that \(\sum_{n=0}^{\infty }\|Q(n)\|e^{\gamma n}<\infty \) for some positive constant \(\gamma .\) The authors firstly study the stability property of the zero solution. Next, the exponential asymptotic stability of the zero solution is analyzed, under some specific hypothesis on the spectrum of the corresponding characteristic operator. This spectrum is characterized with the aid of the Gelfand transform from the theory of commutative Banach algebras. Last part of the paper is devoted to the stability of a positive constant solution of a scalar partial differential equation with piecewise continuous delays. This equation describes the population dynamics in a single species, taking into consideration the diffusion effects.