  We prove new characterisations of exponential stability for positive linear
discrete-time systems in ordered Banach spaces, in terms of small-gain
conditions. Such conditions have played an important role in the
finite-dimensional systems theory, but are relatively unexplored in the
infinite-dimensional setting, yet.
  Our results are applicable to discrete-time systems in ordered Banach spaces
that have a normal and generating positive cone. Moreover, we show that our
stability criteria can be considerably simplified if the cone has non-empty
interior or if the operator under consideration is quasi-compact.
  To place our results into context we include an overview of known stability
criteria for linear (and not necessarily positive) operators and provide full
proofs for several folklore characterizations from this domain.
