## Abstract

A trace formula is proved for pairs of selfadjoint operators that are close to each other in a certain sense. An important role is played by a function analytic in the open upper half-plane and with positive imaginary part there. This function, called the characteristic function of the pair, coincides withKreĭn’s Q-function in the case where the selfadjoint operators are canonical extensions of a common simple and closed Hermitian operator. Special emphasis is given to the finite-dimensional case. Relationships with Kreĭn’s spectral shift function are also considered. Finally, the case of canonical differential expressions is discussed briefly. In this case, the function N may be chosen to be the Weyl function of the canonical differential expression.

Original language | English |
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Pages (from-to) | 59-104 |

Number of pages | 46 |

Journal | St. Petersburg Mathematical Journal |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2004 |

## Keywords

- Kreĭn’s spectral shift function
- The Q-function associated with a symmetric operator
- The Weyl function

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics