A trace formula for isometric pairs

Yang, Rongwei

doi:10.1090/S0002-9939-02-06687-X

A trace formula for isometric pairs
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Proc. Amer. Math. Soc. 131 (2003), 533-541 Request permission

Abstract:

It is well known that for every isometry $V$, $tr[V^{*},\ V]=-ind(V).$ 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 \[ tr[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]=-2ind(V_{1}, V_{2}),\] 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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