# Index and determinant of n-tuples of commuting operators

### Ryszard Nest (University of Copenhagen)

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

#### Abstract

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.