Note on the Kato property of sectorial forms

Abstract

We characterise the Kato property of a sectorial form 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 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 Sp(V) (p⩾1), then the associated form aT(⋅,⋅):=⟨(I+iT)⋅,⋅⟩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 a with respect to H possesses the Kato property.