# Colloquium: Triple operator integrals valued in trace class operators.

### Event Description:

Title: **Triple operator integrals valued in trace class operators.**

Abstract:

In this talk, I will introduce few notions of noncommutative analysis. Among them, I will define important spaces such as the Hilbert-Schmidt class S^2(H) and the trace class S^1(H), where H is a Hilbert space. Those spaces can be seen as the noncommutative analogues of the usual l_2 and l_1-spaces (the spaces of square summable and summable sequences of complex numbers, respectively).

I will also define linear and bilinear *Schur multipliers*. Those objects are related to the Schur products of matrices and to understand them, it is enough to study them in the case of finite matrices.

Finally, I will introduce the *double operator integral mappings* and *triple operator integral mappings* associated to normal operators A,B,C on a Hilbert space H. When H is finite dimensional, those objects correspond to linear and bilinear Schur multipliers with respect to the spectral decompositions of A, B and C. I will present a very recent result concerning them. Also, it turns out that those operator integral mappings have interesting applications to perturbation theory for operators, that is, what could be for instance a *Taylor formula* when we replace real numbers by operators?

This is a joint work with Christian Le Merdy (University of Bourgogne - Franche-Comté) and Fedor Sukochev (UNSW, Sydney).

### Event Contact

**Contact Name: **Anna Skripka