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Necessary conditions for similarity of an operator to a selfadjoint one. (English. Russian original) Zbl 0813.47022
Funct. Anal. Appl. 26, No. 4, 295-297 (1992); translation from Funkts. Anal. Prilozh. 26, No. 4, 80-83 (1992).
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 \(Im\lambda< 0\).
47A65 Structure theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
Full Text: DOI
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