The similarity problem for non-self-adjoint operators with absolutely continuous spectrum.

*(English. Russian original)*Zbl 0985.47017
Funct. Anal. Appl. 34, No. 2, 143-145 (2000); translation from Funkts. Anal. Prilozh. 34, No. 2, 78-81 (2000).

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 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 Jan A. van Casteren, Pac. J. Math. 104, No. 1, 241-255 (1983; Zbl 0457.47002) and 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 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.

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 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 Jan A. van Casteren, Pac. J. Math. 104, No. 1, 241-255 (1983; Zbl 0457.47002) and 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 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.

Reviewer: Daniel Beltita (Bucureşti)

##### Keywords:

absolutely continuous spectrum; non-self-adjoint operator; similarity; functional model; absolute continuity condition
PDF
BibTeX
XML
Cite

\textit{A. V. Kiselev} and \textit{M. M. Faddeev}, Funct. Anal. Appl. 34, No. 2, 143--145 (2000; Zbl 0985.47017); translation from Funkts. Anal. Prilozh. 34, No. 2, 78--81 (2000)

Full Text:
DOI

##### References:

[1] | S. N. Naboko, Funkts. Anal. Prilozhen.,18, No. 1, 16–27 (1984). |

[2] | J. Van Casteren, Pacif. J. Math.,104, No. 1, 241–255 (1983). |

[3] | M. M. Faddeev, Funkts. Anal. Prilozhen.,26, No. 4, 80–83 (1992). |

[4] | B. Sz.-Nagy and C. Foiaş, Analyse harmonique des operateurs de l’espace de Hilbert, Masson, Paris and Akad. Kiadó, Budapest, 1967. · Zbl 0157.43201 |

[5] | B. S. Pavlov, Izv. Akad. Nauk SSSR, Ser. Mat.,39, 123–148 (1975). |

[6] | S. N. Naboko, Trudy Mat. Inst. Steklov.,147, 86–114 (1980). |

[7] | M. S. Brodskii, Triangular and Jordan Representations of Linear Operators, Amer. Math. Soc., Providence, R.I., 1971. |

[8] | L. A. Sakhnovich, Izv. Akad. Nauk SSSR, Ser. Mat.,33, No. 1, 52–64 (1969). |

[9] | V. F. Veselov, Vestn. Leningr. Univ., Ser. 1, No. 4, 19–24 (1988). |

[10] | V. F. Veselov, Vestn. Leningr. Univ., Ser. 1, No. 2, 11–17 (1988). |

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.