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Abstract
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We investigate determinants of Koszul complexes of holomorphic functions of a
commuting tuple of bounded operators acting on a Hilbert space. Our main result
shows that the analytic joint torsion, which compares two such determinants, can be
computed by a local formula which involves a tame symbol of the involved
holomorphic functions. As an application we are able to extend the classical tame
symbol of meromorphic functions on a Riemann surface to the more involved setting
of transversal functions on a complex analytic curve. This follows by spelling out our
main result in the case of Toeplitz operators acting on the Hardy space over the
polydisc.
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Keywords
determinant functors, Koszul complexes, holomorphic
functional calculus, joint torsion, tame symbols
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Mathematical Subject Classification 2010
Primary: 32A10, 47A13
Secondary: 19C20, 19K56, 32C15
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Milestones
Received: 8 February 2018
Revised: 26 July 2019
Accepted: 7 October 2019
Published: 20 June 2020
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