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

Adjoints of elliptic cone operators

“…The main result in that section is Proposition 5.9, a result along the lines of part of the work of Gohberg and Sigal [4]. In Section 6 we prove in a somewhat abstract setting that the pairing alluded to above for solutions with poles at conjugate points is nonsingular (Theorem 6.4).…”

A characterization of generalized poles of generalized Nevanlinna functions

“…Although the invariant subspace L A is not unique, its dimension does not depend on the particular choice. If α is an (ordinary) pole of Q then in general its polar multiplicity (in the sense of [8]) does not coincide with the degree of non-positivity. This is the reason for us not to use the notation multiplicity, that is used e.g.…”

“…In these papers instead of the poles there are treated generalized zeros, but this is essentially the same, since a generalized zero of Q is a generalized pole of the "inverse function" Q(z) := −Q(z) −1 . Pole-cancellation functions in the meromorphic case were introduced in 1971 in [8]. For generalized (that may be non-isolated) poles these functions are discussed in detail in [5].…”

Untitled

“…Here N k is the dimension of the eigenspace of the matrix B αβ (γ k ), φ [22], it follows also that N k n=1 P kn = N • k , in other words, the algebraic multiplicity of the eigenvalue γ k for the matrix B αβ (γ) at γ = γ k is equal to the multiplicity of the zero γ k for the function ∆(γ).…”

“…By (3.4), consequently, B −1 αβ (γ) is a meromorphic function with poles of finite multiplicities in γ k . Moreover, according to [22], Theorem 7.1, the matrix B −1 αβ (γ) has the form…”