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Journal of Operator Theory

Volume 88, Issue 1, Summer 2022  pp. 189-202.

Note on the Kato property of sectorial forms

Authors:  Ralph Chill (1), Sebastian Krol (2)
Author institution: (1) Institut fuer Analysis, Fakultaet Mathematik, Technische Universitaet Dresden, 01062 Dresden, Germany
(2) Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznan, ul. Uniwersytetu Poznanskiego 4, 61-614 Poznan, Poland

Summary:  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 $S_p(V)$ ($p\geqslant 1$), then the associated form $\mathfrak{a}_T(\cdot, \cdot) := \langle(I+\mathrm iT)\cdot,\cdot\rangle_V$ 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.

Keywords: sectorial forms, accretive operators, Kato's square root problem

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