×

zbMATH — the first resource for mathematics

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 \(Im\lambda< 0\).
MSC:
47A65 Structure theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. N. Naboko, Funkts. Anal. Prilozhen.,18, No. 1, 16-27 (1984).
[2] J. Van Casteren, Pacific J. Math.,104, No. 1, 241-255 (1983).
[3] S. N. Naboko, Trudy Mat. Inst. Steklov.,147, 86-114 (1980).
[4] L. A. Sakhnovich, Izv. Akad. Nauk SSSR, Ser. Mat.,33, No. 1, 52-64 (1969).
[5] B. Sz.-Nagy and C. Foia?, Analyse Harmonique des Opérateurs de l’Espace de Hilbert, Masson et Cie ? Akad. Kiadô, Paris?Budapest (1967).
[6] P. Koosis, Introduction toH p spaces, Cambridge Univ. Press, Cambridge (1980). · Zbl 0435.30001
[7] S. N. Naboko, Zap. Nauchn. Sem. LOMI,113, 149-177 (1981).
[8] V. F. Veselov and S. N. Naboko, Mat. Sb.,129 (171, No. 1, 20-39 (1986).
[9] A. V. Straus, Dokl. Akad. Nauk SSSR,126, No. 3, 514-516 (1959).
[10] V. F. Veselov, Vestnik Leningrad. Univ., Math. Mekh. Astron., No. 4, 19-24 (1988).
[11] S. N. Naboko and M. M. Faddeev, Vestnik Leningrad. Univ., Math. Fiz. Khim., No. 4, 78-82 (1990).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.