Linear Analysis Seminar

Date: April 6, 2018

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Dmitriy Zanin, University of New South Wales

Title: Estimates on the singular values for generalised Hilbert transform and double operator integrals

Abstract: If $Hf$ is a Hilbert transform of a function $f,$ then it is well known that $\mu(Hf)\leq (C+C^*)\mu(f),$ where $C$ is the Cesaro operator. This estimate is the best possible. This talk aims to provide a noncommutative analogue of this classical result. The following is demonstrated: if an operator $T$ satisfies $\mathcal{L}_1\to\mathcal{L}_{1,\infty}$ estimate, then $\mu(T(A))\leq (C+C^*)\mu(A).$ In particular, the latter estimate applies to triangular truncation operator (which is considered a noncommutative version of a Hilbert transform). It also applies to certain types of double operator integrals. We show that the estimate above is the best possible for the triangular truncation operator."