Index and determinant of commuting $n$-tuples of operators

Ryszard Nest  (Copenhagen University)

9:30 am to 10:30 am, Mar 26th, 2015   A1510, Science Building

Abstract:

Suppose that $A=(A_1,\ldots A_n )$ is an n-tuple of commuting operators on a Hilbert space and $f=(f_1,\ldots,f_n)$ is an n-tuple of functions holomorphic in a neighbourhood of the (Taylor) spectrum of A. The n-tuple of operators $f(A)=( f_1(A_1,\ldots, A_n),\ldots, f_n(A_1,\ldots, A_n) )$ 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 formulae 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.