Talk by Ralph Chill

On June 16th, 2021, Prof. Dr. Ralph Chill (TU Dresden) gave a talk about "The Kato property of sectorial forms" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.


Show Abstract

We characterise the Kato property of a sectorial form $\mathfrak{a}$, defined on a Hilbert space $\boldsymbol{V}$, with respect to a larger Hilbert space $\boldsymbol{H}$ in terms of two bounded, selfadjoint operators $\boldsymbol{T}$ and $\boldsymbol{Q}$ determined by the imaginary part of $\mathfrak{a}$ and the embedding of $\boldsymbol{V}$ into $\boldsymbol{H}$, respectively.

As a consequence, we show that if a bounded selfadjoint operator $\boldsymbol{T}$ on a Hilbert space $\boldsymbol{V}$ is in the Schatten class $\boldsymbol{S_p (V) (p\geq 1)}$, then the associated form $\boldsymbol{\mathfrak{a}_T(\cdot , \cdot) := \langle (I+iT)\cdot , \cdot \rangle _V }$ has the Kato property with respect to every Hilbert space $\boldsymbol{H}$ into which $\boldsymbol{V}$ is densely and continuously embedded. This result is in a sense sharp. Another result says that if $\boldsymbol{T}$ and $\boldsymbol{Q}$ commute then the form $\mathfrak{a}$ with respect to $\boldsymbol{H}$ possesses the Kato property.

This is joint work with Sebastian Król.

Video of the talk

Liza Schonlau | 10.05.2024