Boundary triplets and maximal accretive extensions of sectorial operators (Q2849105)

A linear operator \(S\) in a complex Hilbert space \(\mathcal{H}\) is called \(\alpha\)-sectorial if its numerical range \(W(S):=\{(Su,u): u\in \operatorname{dom}S,\;\|u\|=1\}\) is contained in the sector \( S(\alpha):=\{z \in \mathbb{C}: | \mathrm{arg} z | \leq \alpha\} \). Such \(S\) is called maximal sectorial (\(m\)-sectorial) if \(S\) is closed and has no accretive extensions in \(\mathcal{H}\). The author of the paper under review presents a detailed survey of results related to the problem of a description of all maximal accretive extensions for densely defined sectorial operators and also \(m\)-sectorial extensions; note that the essential part of the results were obtained by the author individually or with coauthors. All \(m\)-sectorial and \(m\)-accretive extensions by means of boundary pairs and two parameters are described, special boundary triplets allow to determine the actions and domains of extensions. The notions of \(\alpha\)-sectorial relations is used, Weil functions for the case of sectorial operators are considered, analogs of Vishik-Birman-Grubb type formulas are presented in a sectorial setting. Applications to differential operators are also included.NEWLINENEWLINEFor the entire collection see [Zbl 1269.47001].