Reciprocity for Fredholm operators (Lefschetz numbers/Steinberg symbols/holomorphic chains/local index)
The difference between the joint spectrum and the essential joint spectrum of a pair of commuting bounded operators acting on a Hilbert space consists of connected open subsets of \({\mathbb{C}}^ 2\), analytic varieties of complex dimension 1 and isolated points. The authors of the present paper have developed a refined index theory which assigns certain multiplicities even to the above mentioned low dimensional parts of the essential resolvent set, cf. \textit{R. W. Carey} and \textit{J. D. Pincus}: Principal currents, ibid. 8, 614-640 (1985). The present paper fits into the same program. To a pair of Fredholm operators A and B is associated the Fredholm number \[ \psi (B;A)=\det (B| Ker A)/\det (B| Co\ker A). \] The relationship between \(\psi\) (B;A) and \(\psi\) (A;B) is studied from the point of view of algebraic K- theory. Then it is proved that the Lefschetz number preserves the analytic dependence on parameters. With the aid of these tools, the authors investigate the properties of their refined index. Some interesting connections with other areas of mathematics promise a continuation and an interesting evolution of the ideas of Carey and Pincus. However, the paper is intended to mature readers.
essential joint spectrum of a pair of commuting bounded operators acting on a Hilbert space
scientific article; zbMATH DE number 4009127
scientific article