2017/06/24 by Derkach, Vladimir, Hassi, Seppo, Malamud, Mark
#34L40 #35J05 #35J10 #35J25 #46C20 #47A05 #47A10 #47A20 #47A48 #47A57 #47B15 #47B25. Secondary 34B40 #47B36 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 34B08
paper · doi:10.48550/arxiv.1706.07948
With a closed symmetric operator A in a Hilbert space \mathfrak H a triple Π=\\mathcal H,Γ0,Γ1\ of a Hilbert space \mathcal H and two abstract trace operators Γ0 and Γ1 from A^* to \mathcal 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 (essentially) unitary boundary triples which guarantee that the Weyl functions of boundary triples are Nevanlinna functions on \mathcal H, or at least they belong to the class of Nevanlinna families. The boundary condition Γ0f=0 determines a reference operator A0. 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. The case where A0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to PDEs and ODEs. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function M. These characterizations involve solving direct and inverse problems for specific classes of (unbounded) operator functions M. One of the main results specifies the analytic properties of M which guarantee that A0 is essentially selfadjoint. In this study we also derive, for instance, Kre\uın-type resolvent formulas for the most general classes of unitary and isometric boundary triples appearing in the present work. All the main results are shown to have applications in the study of ordinary and partial differential operators.