An Operator Generalization of the Logarithmic Residue Theorem and the Theorem of Rouché

We obtain the operator generalization of the theorem on the logarithmic residue for meromorphic operator-functions. The proof of the generalization is based on a theorem concerning a special factorization of a meromorphic operator-function at a point. This theorem also allows us to generalize, to the case of meromorphic operator-functions, the formula of M. V. Keldyš for the principal part of the resolvent as well as several other theorems. A definition is given for the multiplicity of a pole for a meromorphic operator-function. The basic properties of the multiplicity of a pole are proved, and also a generalization of the Rouché theorem. Bibliography: 16 items.