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. Keldysh 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 titles.