THE SPECTRUM AND EIGENSPACES OF A MEROMORPHIC OPERATOR-VALUED FUNCTION

It is shown how to associate eigenvectors with a meromorphic mapping d efined on a Riemann surface with values in the algebra of bounded oper ators on a Banach space. This generalises the case of classical spectr al theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which a sserts the completeness of the eigenvectors whenever the Riemann surfa ce is compact. Two technical tools are discussed in detail: Cauchy-ker nels and Runge's Approximation Theorem for vector-valued functions.