Note on the Kato property of sectorial forms

TL;DR

This paper characterizes the Kato property of sectorial forms on Hilbert spaces using bounded operators, establishing conditions under which the property holds, including Schatten class and commutativity cases, with implications for operator theory.

Contribution

It provides a new characterization of the Kato property for sectorial forms via bounded operators, extending understanding of when the property holds in various operator classes.

Findings

If T is in Schatten class S_p, the form has the Kato property for all embeddings.

Commuting T and Q imply the form has the Kato property.

The results are sharp and extend previous knowledge on sectorial forms.

Abstract

We characterise the Kato property of a sectorial form $\mathfrak{a}$, defined on a Hilbert space $V$, with respect to a larger Hilbert space $H$ in terms of two bounded, selfadjoint operators $T$ and $Q$ determined by the imaginary part of $\mathfrak{a}$ and the embedding of $V$ into $H$, respectively. As a consequence, we show that if a bounded selfadjoint operator $T$ on a Hilbert space $V$ is in the Schatten class $S_p(V)$ ($p\geq 1$), then the associated form $\mathfrak{a}_T(\cdot, \cdot) := \langle (I+iT)\cdot ,\cdot\rangle_V$ has the Kato property with respect to every Hilbert space $H$ into which $V$ is densely and continuously embedded. This result is in a sense sharp. Another result says that if $T$ and $Q$ commute then the form $\mathfrak{a}$ with respect to $H$ possesses the Kato property.