During the last few years the authors have studied extensively the invarian
t subspace problem of positive operators; see [6] for a survey of this inve
stigation. In [4] the authors introduced the class of compact-friendly oper
ators and proved for them a general theorem on the existence of invariant s
ubspaces. It was then asked if every positive operator is compact-friendly
In this note, we present an example of a positive operator which is not com
pact-friendly but which, nevertheless, has a non-trivial closed invariant s
ubspace.
In the process of presenting this example, we also characterize the multipl
ication operators that commute with non-zero finite-rank operators. We show
, among other things, that a multiplication operator M phi commutes with a
non-zero finite-rank operator if and only the multiplier function cp is con
stant on some non-empty open set.