article

Title: Operators with a compact imaginary part

Authors: C. Foias, S. Hamid, C. Onica and C. Pearcy

Issue: Volume 58 (2009), Issue 5, 2297-2304

Abstract:

$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.$