Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions

With a closed symmetric operator A in a Hilbert space (Formula presented.) a triple (Formula presented.) of a Hilbert space (Formula presented.) and two abstract trace operators Γ0 and Γ1 from (Formula presented.) to (Formula presented.) is called a generalized boundary triple for (Formula presented.) if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions (Formula presented.) are investigated. The most important ones for applications are specific classes of boundary triples for which Green's second identity admits a certain maximality property which guarantees that the corresponding Weyl functions are Nevanlinna functions on (Formula presented.), i.e. (Formula presented.), or at least they belong to the class (Formula presented.) of Nevanlinna families on (Formula presented.). The boundary condition (Formula presented.) determines a reference operator (Formula presented.). The case where A0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ0 and Γ1 admits a von Neumann type decomposition via A0 and the defect subspaces of A. The case where A0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to boundary value problems e.g. in PDE setting or when modeling differential operators with point interactions. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function (Formula presented.) and close interconnections between different classes of boundary triples and the corresponding transformed/renormalized Weyl functions are investigated. These characterizations involve solving direct and inverse problems for specific classes of operator functions (Formula presented.). Most involved ones concern operator functions (Formula presented.) for which (Formula presented.) defines a closable nonnegative form on (Formula presented.). It turns out that closability of (Formula presented.) does not depend on (Formula presented.) and, moreover, that the closure then is a form domain invariant holomorphic function on (Formula presented.) while (Formula presented.) itself need not be domain invariant. In this study we also derive several additional new results, for instance, Kreĭn-type resolvent formulas are extended to the most general setting of unitary and isometric boundary triples appearing in the present work. In part II of the present work all the main results are shown to have applications in the study of ordinary and partial differential operators. © 2020 The Authors. Mathematische Nachrichten published by Wiley-VCH Verlag GmbH & Co. KGaA