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=(A_{1},..., A_{n})
is an n-tuple of commuting
operators on a Hilbert space and f=(f_{1},..., 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},..., A_{n}),...,
f_{n}(A_{1},..., A_{n}))
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.