Talk by Ralph Chill

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.