To each semibounded form ${\mathfrak t}$ in a Hilbert space ${\mathfrak H}$, not necessarily closed or closable, with lower bound $\gamma\in {\mathbb R}$ one can associate a representing map $Q$ from ${\mathfrak H}$ to an auxiliary Hilbert space ${\mathfrak K}$ via
$${\mathfrak t}[\varphi, \psi]=c(\varphi, \psi)+(Q\varphi, Q\psi), \quad \varphi, \psi \in {\rm dom}\, {\mathfrak t}, $$
for some $c \leq \gamma$.
Representing maps are not uniquely determined. If, in addition, $Q$ has dense range in ${\mathfrak K}$, then $Q$ becomes essentially unique (up to unitary equivalence). In this talk the power of representing maps for the study of forms is shown via a couple of applications. For instance, all Lebesgue type decomposition of the form ${\mathfrak t}={\mathfrak t}_1+{\mathfrak t}_2$, where the form ${\mathfrak t}_1$ is closable and the form ${\mathfrak t}_2$ is singular, are described explicitly by means of $Q$. Also the interconnection between ${\mathfrak t}_1$ and ${\mathfrak t}_2$ is studied.
A representing map $Q$ for the form ${\mathfrak t}$ generates a semibounded symmetric relation $S_{\mathfrak t}=Q^*Q+c$ and a selfadjoint semibounded relation $\widetilde{A}_{\mathfrak t}=Q^*Q^{**} +c$, which are uniquely determined by the form ${\mathfrak t}$ itself. Characterizations of $S_{\mathfrak t}$ and $\widetilde{A}_{\mathfrak t}$ both generalize the first representation theorem, which is well-known in the case of closed forms. In the densely defined case $\widetilde{A}_{\mathfrak t}$ coincides with the selfadjoint operator associated with the real part of a sectorial form which in the case of nonclosable forms has been studied by W. Arendt and T. ter Elst.
The talk is based on joint work with Henk de Snoo.