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

Volodymyr Derkach, Seppo Hassi and Mark Malamud

Mathematische Nachrichten, 2020, vol. 293, issue 7, 1278-1327

Abstract: With a closed symmetric operator A in a Hilbert space H a triple Π={H,Γ0,Γ1} of a Hilbert space H and two abstract trace operators Γ0 and Γ1 from A∗ to H is called a generalized boundary triple for A∗ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions M(·) 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 H, i.e. M(·)∈R(H), or at least they belong to the class R∼(H) of Nevanlinna families on H. The boundary condition Γ0f=0 determines a reference operator A0(=kerΓ0). 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 M(·) 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 M(·). Most involved ones concern operator functions M(·)∈R(H) for which τM(λ)(f,g)=(2iImλ)−1[(M(λ)f,g)−(f,M(λ)g)],f,g∈domM(λ),defines a closable nonnegative form on H. It turns out that closability of τM(λ)(f,g) does not depend on λ∈C± and, moreover, that the closure then is a form domain invariant holomorphic function on C± while τM(λ)(f,g) 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.

Date: 2020
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