Representing maps for semibounded forms and their applications

Seppo Hassi (University of Vaasa)

Nov 10. 2022, 09:45 — 10:15

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.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Spectral Theory of Differential Operators in Quantum Theory (Workshop)
Jussi Behrndt (TU Graz)
Fritz Gesztesy (Baylor U, Waco)
Ari Laptev (Imperial College London)
Christiane Tretter (U Bern)