Mathematics Colloquium

Albrecht Pietsch


"Traces of operators and their history"

Date: Mon, October 14, 2013
Time: 17:15
Place: Research II Lecture Hall

Abstract: As every mathematician knows, the trace of a square matrix is defined to be the sum of all entries of the main diagonal. Extending this concept to the infinite-dimensional setting does not always work, since non-converging infinite series may occur. So one had to identify those operators that possess something like a trace. In a first step, this was done for operators on the separable Hilbert space. The situation in Banach spaces turned out to be much more complicated, as the missing approximation property causes a lot of trouble. I will present an axiomatic approach in which operator ideals play a dominant rule. My considerations include also singular traces that -by definition- vanish on all finite rank operators. Thanks to the discoveries of Connes, those traces became a useful tool in non-commutative geometry, in the theory of pseudo-differential operators, and in quantum mechanics. The lecture is intended to show that people who prefer to live in Hilbert spaces heavily need Banach spaces techniques.

S. Lord, F. Sukochev, and D. Zanin, Singular Traces, De Gruyter, Berlin, 2012.
A. Pietsch, History of Banach Spaces and Linear Operators, Birkhauser, Boston, a 2007.
A. Pietsch, Traces on operator ideals and related linear forms on sequence ideals (part I ), to appear in Indag. Math., DOI: 10.1016/j.indag.2012.08.008.