---
abstract: 'Let $T$ be a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$ and let $B \in \mathbf{B}(\sK)$ be selfadjoint. It will be shown that the relation $T^{*}(I+iB)T$ is maximal sectorial via a matrix decomposition of $B$ with respect to the orthogonal decomposition $\sH=\cdom T^* \oplus \mul T$. This leads to an explicit expression of the corresponding closed sectorial form. These results include the case where $\mul T$ is invariant under $B$. The more general description makes it possible to give an expression for the extremal maximal sectorial extensions of the sum of sectorial relations. In particular, one can characterize when the form sum extension is extremal.'
address:
- |
     Department of Mathematics and Statistics\
    University of Vaasa\
    P.O. Box 700, 65101 Vaasa\
    Finland
- |
     Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence\
    University of Groningen\
    P.O. Box 407, 9700 AK Groningen\
    Nederland
author:
- 'S. Hassi'
- 'H.S.V. de Snoo'
title: A class of sectorial relations and the associated closed forms
---

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Introduction
============

A linear relation $H$ in a Hilbert space $\sH$ is said to be *accretive* if $\RE (h',h) \geq 0$, $\{h,h'\} \in H$. Note that the closure 