Representing maps for semibounded forms and their Lebesgue type 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 ${\mathfrak t}[\varphi, \psi]-c(\varphi, \psi)=(Q\varphi,Q\psi)$, $\varphi,\psi \in {\rm dom\,}{\mathfrak t}$, 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.