  We study problems associated with an operator pencil, i.e., a pair of
operators on Banach spaces. Two natural problems to consider are linear
constrained differential equations and the description of the generalized
spectrum. The main tool to tackle either of those problems is the reduction of
the pencil. There are two kinds of natural reduction operations associated to a
pencil, which are conjugate to each other.
  Our main result is that those two kinds of reductions commute, under some
mild assumptions that we investigate thoroughly.
  Each reduction exhibits moreover a pivot operator. The invertibility of all
the pivot operators of all possible successive reductions corresponds to the
notion of regular pencil in the finite dimensional case, and to the inf-sup
condition for saddle point problems on Hilbert spaces.
  Finally, we show how to use the reduction and the pivot operators to describe
the generalized spectrum of the pencil.
