Boundary triplets and Weyl functions. Recent developments (Q2849109)

A theory of extensions of Hilbert space symmetric operators, based on the concepts of boundary triplets and associated Weyl functions, was developed by the authors in [Methods Funct. Anal. Topol. 6, No. 3, 24--55 (2000; Zbl 0973.47020); J. Math. Sci., New York 73, No. 2, 141--242 (1995; Zbl 0848.47004); J. Funct. Anal. 95, No. 1, 1--95 (1991; Zbl 0748.47004)], \textit{V. I. Gorbachuk} and \textit{M. L. Gorbachuk} [Boundary value problems for operator differential equations. Dordrecht etc.: Kluwer (1991; Zbl 0751.47025)] and later in [\textit{V. Derkach} et al., Trans. Am. Math. Soc. 358, No. 12, 5351--5400 (2006; Zbl 1123.47004)], where that theory was extended to symmetric relations rather than operators.NEWLINENEWLINEThe paper under review describes different steps in the development of boundary triplets, from ordinary boundary triplets to boundary relations. Starting with a short introduction to linear relations in Hilbert and in Kreĭn spaces, the authors discuss Calkin's approach by means of reduction operators, present necessary facts about ordinary and generalized boundary triplets, and characterize the spectrum of proper extensions in terms of the boundary operator and Weyl functions. Infinite dimensional restrictions of selfadjoint relations are introduced and considered as general models for generalized boundary triplets of bounded type. Special cases of the so-called unitary boundary triplets are considered, and isometric boundary triplets are introduced and studied. Useful applications and examples are included. A detailed list of the contemporary literature on the subject is included in the References.NEWLINENEWLINEFor the entire collection see [Zbl 1269.47001].