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 H a triple Π = { H , Γ 0 , Γ 1 } of a Hilbert space H and two abstract trace operators Γ 0 and Γ 1 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 classR ∼ ( H )of Nevanlinna families on H . The boundary conditionΓ 0f = 0 determines a reference operatorA 0 ( =kerΓ 0 ) . The case where A 0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ 0 and Γ 1 admits a von Neumann type decomposition via A 0 and the defect subspaces of A . The case where A 0 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 whichτ M ( λ )( f , g ) = ( 2 iImλ ) − 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 ofτ M ( λ )( f , g )does not depend on λ ∈ C ±and, moreover, that the closure then is a form domain invariant holomorphic function on C ± whileτ M ( 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.