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We propose a method of constructing examples in operator ergodic theory which unifies and extends some previously known examples. It also allows us to answer several questions that have been open for some time (including a question of Allan ).
A simple geometrical argument shows that every pair of projections
on a finite-dimensional complex
vector space has a common invariant subspace of dimension 1 or 2. The idea
extends to certain pairs of
projections on an infinite-dimensional Hilbert space H. In particular
every projection on H has a reducing
subspace, although a finite-dimensional one need not exist. In a final
section, the results are extended to
the existence of hyperinvariant subspaces for pairs of projections.
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