### Mathematics Colloquium

# Albrecht Pietsch

### (Jena)

## "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.

REFERENCES:

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.