Natural Representations of the Multiplicity of an Analytic Operator-valued Function at an Isolated Point of the Spectrum

Abstract.

Representations are given for the multiplicity of an analytic operator-valued function A at an isolated point z₀ of the spectrum in the form of kernels and ranges of Hankel and Toeplitz matrices whose entries are derived from the Taylor coefficients of A and the Laurent coefficients of A⁻¹ about z₀. In two special cases the results can be expressed in terms of finite matrices: when A is a polynomial and when A⁻¹ has a pole at z₀. The latter case leads to the theory of Jordan chains.