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Factorized sectorial relations, their maximal-sectorial extensions, and form sums. (English) Zbl 1423.47011

Summary: In this paper, we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space \(H\). Our 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 self-adjoint operator in a Hilbert space \(\mathfrak{K}\) and \(T:\mathfrak{H}\rightarrow\mathfrak{K}\) (or \(T:\mathfrak{K}\rightarrow\mathfrak{H}\), 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.

MSC:

47B44 Linear accretive operators, dissipative operators, etc.
47A06 Linear relations (multivalued linear operators)
47A07 Forms (bilinear, sesquilinear, multilinear)
47B65 Positive linear operators and order-bounded operators
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