Index and determinant of n-tuples of commuting operators

Ryszard Nest (University of Copenhagen)

Friday 29th November, 2013 17:00-17:55 516


Suppose that $ A = (A_1, \dots A_n ) $ is an n-tuple of commuting operators on a Hilbert space and $ f = (f_1, \dots,f_n) $ is an n-tuple of functions holomorphic in a neighbourhood of the (Taylor) spectrum of $ A $. The n-tuple of operators $ f(A) = (f_1(A_1, \dots, A_n), \dots, f_n(A_1, \dots, A_n) ) $ give rise to a complex $ {\mathcal K}(f(A),H) $, its so called Koszul complex, which is Fredholm whenever $ f^{-1}(0) $ does not intersect the essential spectrum of $ A $. Given that $ f $ satisfies the above condition, we will give a local formula for the index and determinant of $ {\mathcal K}(f(A),H) $. The index formula is a generalisation of the fact that the winding number of a continuous nowhere zero function $ f $ on the unit circle is, in the case when it has a holomorphic extension $ tilde{f} $ to the interior of the disc, equal to the number of zero's of $ tilde{f} $ counted with multiplicity.

The explicit local formula for the determinant of $ {\mathcal K}(f(A),H) $ can be seen as an extension of the Tate tame symbol to, in general, singular complex curves.

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