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A trace formula for isometric pairs

Authors:	Rongwei Yang

Affiliation:	Department of Mathematics, Arizona State University, Tempe, Arizona 85287

Abstract:	It is well known that for every isometry , $tr[V^{}, V]=-ind(V).$ This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries $(V_{1}, V_{2})$ , namely $begin{displaymath}tr[V_{1}^{},V_{1},V_{2}^{},V_{2}]=-2ind(V_{1}, V_{2}),end{displaymath}$ where $[V_{1}^{},V_{1},V_{2}^{},V_{2}]$ is the complete anti-symmetric sum and $ind(V_{1}, V_{2})$ is the Fredholm index of the pair $(V_{1}, V_{2})$ . The major tool is what we call the fringe* operator. Two examples are considered.

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