Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) łp-input łq-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to łq when non-homogeneous terms are in
Nearby in the stack
łp
. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted
łr
-space with an exponentially fading weight (the phase space). Our main result states that (i)
⇔
(ii) whenever
(p,q)=(1,∞)
and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and
łp
-input
łq
-state stabilities does not depend on the choice of a phase space and parameters
p
and
q
, respectively.
ł1
-input
ł∞
-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.