Basic representation theorems of forms
Zoltán Sebestyén, Zsigmond Tarcsay
TL;DR
The paper extends Kato’s representation framework to densely defined nonnegative sesquilinear forms that need not be closed or closable, by constructing a maximal representing operator $T$ with $\mathcal{D_*} = \operatorname{dom} T \cap \mathcal{D}$ and $\mathfrak{t}(g,h) = \langle Tg, h \rangle$ for $g \in \mathcal{D_*}, h \in \mathcal{D}$, while also obtaining $\mathfrak{t}(g,h) = \langle T^{1/2}g, T^{1/2}h\rangle$ on $\mathcal{D_{**}}$. The work introduces a quotient-embedding construction $Q$ and shows $T = J^{**}J^*$, establishing a maximality property with respect to the form-order $\preceq$ and linking to Kato’s first and second representation theorems. In addition, the authors present a concise proof that the Friedrichs extension of a densely defined positive operator is the positive self-adjoint extension $T_F = Q^* Q^{**}$, and they characterize the closability and boundedness of the form via $\mathcal{D_*}$ and $\mathcal{D_{**}}$, including the case $\mathcal{D_*} = \mathcal{D}$ where $T_F$ coincides with the closure. Overall, the results hold in both real and complex Hilbert spaces and provide a natural, minimalistic framework for associating positive self-adjoint operators with broader classes of forms.
Abstract
We study maximal representations of nonnegative sesquilinear forms in real or complex Hilbert spaces, that are not necessarily closed or even closable. We associate positive self-adjoint operators with such forms, in a sense similar to Kato's representation theorems. In particular, we give a brief proof of the Friedrichs extension of a densely defined positive operator.