The Friedrichs Extension of an Unbounded Operator

Definition 1. Let \(\mathcal{H}_0\) and \(\mathcal{H}_1\) be Hilbert spaces. An unbounded operator \(\mathcal{H}_0 \to \mathcal{H}_1\) is a linear map

\begin{equation*} T ~\colon \mathrm{dom}(T) \longrightarrow \mathcal{H}_1 \end{equation*}

where \(\mathrm{dom}(T) \subseteq \mathcal{H}_0\) is required to be a dense subspace. Since an operator of this form can be constructed by simply restricting a bounded operator on \(\mathcal{H}_0\) to \(\mathrm{dom}(T)\), we also require that \(T\) not be such a restriction. ∎

Definition 2. Let \(T\) and \(S\) are unbounded operators \(\mathcal{H}_0 \to \mathcal{H}_1\). If \(\mathrm{dom}(T) \subseteq \mathrm{dom}(S)\) and \(S| _ {\mathrm{dom}(T)} \equiv T\) then we say that \(S\) is an extension of \(T\). For short we often write this as \(T \subset S\).∎

Definition 3. Let \(T \colon \mathcal{H}_0 \to \mathcal{H}_1\) and \(S \colon \mathcal{H}_1 \to \mathcal{H}_0\) be unbounded operators. We say that \(S\) a sub-adjoint to \(T\) if, for all \(v \in \mathrm{dom}(T)\) and \(w \in \mathrm{dom}(S)\), we have

\begin{equation*} \langle Tv, w \rangle = \langle v, Sw \rangle \end{equation*}

∎

If \(\mathcal{H}_0\) and \(\mathcal{H}_1\) are Hilbert spaces, then the direct sum \(\mathcal{H}_0 \oplus \mathcal{H}_1\) is also a Hilbert space. The inner product on the direct sum is given by

\begin{equation*} \langle v_0 \oplus v_1 , w_0 \oplus w_1 \rangle = \langle v_0, w_0 \rangle + \langle v_1, w_1 \rangle \end{equation*}