## Banach Journal of Mathematical Analysis

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

#### Abstract

In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space ${\mathfrak{H}}$. Our particular interest is in sectorial relations $S$, which can be expressed in the factorized form $\begin{equation*}S=T^{*}(I+iB)T\qquad \text{or}\qquad S=T(I+iB)T^{*},\end{equation*}$ where $B$ is a bounded self-adjoint 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, along with a straightforward construction of the extreme extensions $S_{F}$, the Friedrichs extension, and $S_{K}$, the Kreĭn extension of $S$, which uses the above factorized form of $S$. As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 3 (2019), 538-564.

Dates
Accepted: 28 January 2019
First available in Project Euclid: 25 May 2019

https://projecteuclid.org/euclid.bjma/1558749978

Digital Object Identifier
doi:10.1215/17358787-2019-0003

Mathematical Reviews number (MathSciNet)
MR3978936

Zentralblatt MATH identifier
07083760

#### Citation

Hassi, Seppo; Sandovici, Adrian; de Snoo, Henk. Factorized sectorial relations, their maximal-sectorial extensions, and form sums. Banach J. Math. Anal. 13 (2019), no. 3, 538--564. doi:10.1215/17358787-2019-0003. https://projecteuclid.org/euclid.bjma/1558749978

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