correspond to $H$; cf. Theorem \[s-first\]. Since the closed form $\st_H$ is sectorial, one has the inequality $$\label{henillla0}
 |(\st_{H})_{\rm i}[h] | \leq (\tan \alpha) (\st_{H})_{\rm r}[h], \quad h \in \dom \st,$$ and in this situation the real part $(\st_{H})_{\rm r}$ is a closed nonnegative form. Hence by the first representation theorem there exists a nonnegative selfadjoint relation $H_{\rm r}$, the so-called *real part* of $H$, such that $\dom H_{\rm r} \subset \dom (\st_{H})_{\rm r}=\dom \st_{H}$ and $$(\st_{H})_{\rm r}[h,k] = ( h', k ), \quad \{ h,h' \} \in H_{\rm r},
\quad k \in \dom (\st_{H})_{\rm r}=\dom \st_{H}.$$This real part $H_{\rm r}$, not to be confused with the real part introduced in [@HSnSz09], will play an important role in formulating the second representation theorem below. First the case where $H$ is a maximal sectorial operator will be considered, in which case $H$ is automatically densely defined; see [@Kato].

\[s-thirdLemma\] Let $H$ be an maximal sectorial operator in $\sH$, let the closed sectorial form $\st_H$ correspond to $H$ via Theorem \[s-first\], and let $H_{\rm r}$ be the real part of $H$. Then there exists a unique selfadjoint operator $G \in \mathbf{B}(\sH)$ with $\| G \| = \tan \alpha$, of the form $$\label{Bdec1++}
 