Necessary conditions for similarity of an operator to a selfadjoint one (Q1320684)

Let \(L\) be an operator in a Hilbert space \(H\). \(L\) is called similar to a self-adjoint operator if there exist a selfadjoint operator \(A\) and a bounded operator \(X\) (with bounded inverse in \(H\)) such that \(L= X^{-1} AX\). The paper gives several necessary and sufficient conditions for \(L\) to be similar to a selfadjoint operator, and discusses also some necessary, but not sufficient conditions. The results are mostly very technical and cannot be reproduced here. But we would like to mention the following result for an operator of the form \(L= A+ V\), where \(V\) is a rank-one operator. If an operator \(L\) with rank \(ImL\leq 2\) is similar to a self-adjoint operator, then the determinant of its characteristic function is bounded in the half-planes \(Im\lambda> 0\) and \(Im\lambda< 0\).