Abstract
Let L(λ) be an analytic matrix function, let F be a bounded and isolated part of its numerical range, and Γ be a closed contour of regular points of L(λ) with F inside Γ. Defining c(F) = indΓ (L(λ) f, f), f≠ 0, we give new proofs of the existence of spectral divisors of L(λ.) with respect toΓin the cases that either c(F) = 1 or Γ is a circle and, in both cases, there is no completeness hypothesis. Other results on the existence of Γ-spectral divisors (using completeness) are extended from the case of matrix polynomials to that of analytic matrix functions.
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