Speaker: Ryszard Nest (Univ. Copenhagen)

Title: Index and determnant of n-tuples of commuting operators

Time/Date: 4:30-6:00pm, Wednesday, April 9, 2014

Room: 122 Math. Sci. Building

Abstract: Suppose that A=(A1,..., An) is an n-tuple of commuting operators on a Hilbert space and f=(f1,..., fn) is an n-tuple of functions holomorphic in a neighbourhood of the (Taylor) spectrum of A. The n-tuple of operators f(A)=(f1(A1,..., An),..., fn(A1,..., An)) give rise to a complex 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 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 f˜ to the interior of the disc, equal to the number of zero's of f˜ counted with multiplicity. The explicit local formula for the determinant of K(f(A),H) can be seen as an extension of the Tate tame symbol to, in general, singular complex curves.