Operators with a compact imaginary part Ciprian FoiasSami HamidConstantin OnicaCarl Pearcy 47A15invariant subspacescompact imaginary part In this note we initiate a study of the old unsolved problem whether every $T \in \mathcal{L}(\mathcal{H})$ of the form $T = H + iK$ with $K$ compact has a nontrivial invariant subspace, using [C. Foias, C. Pasnicu, and D. Voiculescu, \emph{Weak limits of almost invariant projections}, J. Operator Theory \textbf{2} (1979), 79--93] as our main tool. In case $K \geq 0$ we obtain some positive results. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3963 10.1512/iumj.2009.58.3963 en Indiana Univ. Math. J. 58 (2009) 2297 - 2304 state-of-the-art mathematics http://iumj.org/access/