# Factorized sectorial relations, their maximal sectorial extensions, and form sums

## TL;DR

This paper studies sectorial relations in Hilbert spaces, providing a new factorized form approach to describe their maximal extensions and form sums, with explicit constructions of Friedrichs and Kre
29n extensions.

## Contribution

It introduces a novel factorized form for sectorial relations to explicitly characterize all maximal sectorial extensions and their form sums.

## Key findings

- Explicit description of all maximal sectorial extensions using factorized form.
- Construction methods for Friedrichs and Kre
29n extensions.
- Application to form sums of maximal sectorial extensions.

## Abstract

In this paper sectorial operators, or more generally, sectorial relations and their maximal sectorial extensions in a Hilbert space ${\mathfrak H}$ are considered. The particular interest is in sectorial relations $S$, which can be expressed in the factorized form \[ S=T^*(I+iB)T \quad \text{or} \quad S=T(I+iB)T^*, \] where $B$ is a bounded selfadjoint operator in a Hilbert space ${\mathfrak K}$ and $T:{\mathfrak H}\to{\mathfrak K}$ or $T:{\mathfrak K}\to{\mathfrak H}$, respectively, is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of $S$, a description of all the maximal sectorial extensions of $S$ is given with a straightforward construction of the extreme extensions $S_F$, the Friedrichs extension, and $S_K$, the Kre\u{\i}n extension of $S$, which uses the above factorized form of $S$. As an application of this construction the form sum of maximal sectorial extensions of two sectorial relations is treated.