Pairs of selfadjoint operators and their invariants

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 with Kreĭn'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.