  It is known that hyperbolic linear delay difference equations are shadowable
on the half-line. In this paper, we prove the converse and hence the
equivalence between hyperbolicity and the positive shadowing property for the
following two classes of linear delay difference equations: (a)~for
nonautonomous equations with finite delays and uniformly bounded compact
coefficient operators in (possibly infinite-dimensional) Banach spaces, (b)~for
Volterra difference equations with infinite delay in finite dimensional spaces.
