The similarity problem for non-self-adjoint operators with absolutely continuous spectrum (Q1580173)

Let \(L=A+iV\) be a non-self-adjoint operator in a Hilbert space \(H\), where \(A\) is a self-adjoint operator and \(V\) is an \(A\)-bounded symmetric operator with relative bound less than \(1\), and suppose that the spectrum of \(L\) is real. The paper under review gives some necessary and sufficient conditions for \(L\) to be similar to a self-adjoint operator, within the hypothesis that the spectrum of \(L\) satisfies an absolute continuity condition [which was introduced by \textit{S. N. Naboko}, Tr. Mat. Inst. Steklova 147, 86-114 (Russian) (1980; Zbl 0445.47010)]. These criteria are formulated in terms of the properties of the boundary values on the real axis of the resolvent of \(L\); and they are variants, for the case where \(L\) has absolutely continuous spectrum, of the following result: \(L\) is similar to a self-adjoint operator if and only if \[ \sup_{\varepsilon>0}\varepsilon\int_{\mathbb R} |(L-k-i\varepsilon)^{-1}u\|^2 dk\leq C\|u\|^2 \text{ for every }u\in H \] [see \textit{Jan A. van Casteren}, Pac. J. Math. 104, No. 1, 241-255 (1983; Zbl 0457.47002) and \textit{S. N. Naboko}, Funct. Anal. Appl. 18, 13-22 (1984); translation from Funkts. Anal. Prilozh. 18, No. 1, 16-27 (1984; Zbl 0551.47012)]. One can find the prelimit form of these criteria in a paper by \textit{M. M. Malamud} [Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270, Issled. po Linein. Oper. i Teor. Funkts. 28, 201-241, 367 (2000)] for operators which are not necessarily absolutely continuous.