Representing maps for semibounded forms and their Lebesgue type decompositions
Seppo Hassi, Henk de Snoo
TL;DR
The paper develops a unified operator-theoretic framework for semibounded sesquilinear forms by introducing representing maps ${Q}$ for ${\mathfrak t}-c$ and translating form-theoretic properties (closability, closedness, singularity) into properties of these maps. It establishes a Lebesgue-type decomposition of semibounded forms ${\mathfrak t}={\mathfrak t}_{\rm reg}+{\mathfrak t}_{\rm sing}$, where the regular part is closable and the singular part is nonnegative and singular, with ${\mathfrak t}_{\rm reg}$ being the largest closable form below ${\mathfrak t}$; this decomposition is obtained via the Lebesgue decomposition of the representing map $Q$. The paper then analyzes sum decompositions of nonnegative forms through nonnegative contractions $K$, providing a precise parametrization and mutual singularity criteria in terms of $K$, and introducing the parallel sum as a measure of interaction. Extending these ideas to semibounded forms, Lebesgue-type decompositions are characterized by contractions satisfying specific regular/singular compatibility conditions, connecting to orthogonal projections and mutual singularity. Representation theorems relate semibounded forms to semibounded selfadjoint relations $S_{\mathfrak t}$ and ${\widetilde A}_{\mathfrak t}$ via $S_{\mathfrak t}=Q_c^{*}Q_c+c$ and ${\widetilde A}_{\mathfrak t}=Q_c^{*}Q_c^{**}+c$, yielding a limiting interpretation of the classical representation theorem in a sectorial-type setting and clarifying the regular part as the operator-theoretic core. Finally, monotone (increasing or decreasing) sequences of semibounded forms are shown to converge in the appropriate operator-topology, with corresponding convergence of the regular parts and of the associated selfadjoint relations, enabling a coherent treatment of limits of form families and their decompositions.
Abstract
For a semibounded sesquilinear form ${\mathfrak t}$ in a Hilbert space ${\mathfrak H}$ there exists a representing map $Q$ from ${\mathfrak H}$ to another Hilbert space ${\mathfrak K}$, such that ${\mathfrak t}[\varphi, ψ]-c(\varphi, ψ)=(Q\varphi,Qψ)$, $\varphi,ψ\in {\rm dom\,}{\mathfrak t}$, with $c \in {\mathbb R}$ a lower bound of ${\mathfrak t}$. Representing maps offer a simplifying tool to study general semibounded forms. By means of representing maps closedness, closability, and singularity of ${\mathfrak t}$ are immediately translated into the corresponding properties of the operator $Q$, and vice versa. Also properties of sum decompositions ${\mathfrak t}={\mathfrak t}_1+{\mathfrak t}_2$ of a nonnegative form ${\mathfrak t}$ with two other nonnegative forms ${\mathfrak t}_1$ and ${\mathfrak t}_2$ in ${\mathfrak H}$ can be analyzed by means of associated nonnegative contractions $K\in {\mathbf B}({\mathfrak K})$. This helps, for instance, to establish an explicit operator theoretic characterization for the summands ${\mathfrak t}_1$ and ${\mathfrak t}_2$ to be, or not to be, mutually singular. Such sum decompositions are used to study characteristic properties of the so-called Lebesgue type decompositions of semibounded forms ${\mathfrak t}$, where ${\mathfrak t}_1$ is closable and ${\mathfrak t}_2$ singular; in particular, this includes the Lebesgue decomposition of a semibounded form due to B. Simon. Furthermore, for a semibounded form ${\mathfrak t}$ with its representing map $Q$ it will be shown that the corresponding semibounded selfadjoint relation $Q^*Q^{**} +c$ is uniquely determined by a limit version of the classical representation theorem for the form ${\mathfrak t}$, being studied by W. Arendt and T. ter Elst in a sectorial context. Via representing maps a full treatment is given of the convergence of monotone sequences of semibounded forms.