Boundary quadruples and bijective realisations of abstract Friedrichs operators

Abstract

The theory of boundary quadruples and boundary triples is well-studied for symmetric and skew-symmetric operators and in general for dual-pairs. This paper adapts a suitable version for abstract Friedrichs operators and addresses the following questions: which parameters yield bijective realisations, and which parameters yield $m-$accretive realisations. We study a boundary-quadruple framework in which closed realisations are parametrised by closed relations in a boundary space. This yields the intrinsic criterion \[T_Θ \rm{is} \ \rm{bijective}\iff \lK=Θ\dotplus Γ(\ker T_1) \;.\] For bounded operator parameters $ϕ:\lK_1\to \lK_0$ in the boundary space, we introduce the reference operator \[Q_0=Γ_1(Γ_0|_{\ker T_1})^{-1}\,,\] prove that $\|Q_0\|< 1$, and obtain the exact criterion \[T_ϕ\ \rm{is} \ \rm{bijective} \iff \I_{\lK_0}-ϕQ_0\ \rm{is} \ \rm{bijective}\;.\] Consequently, every non-expansive parameter gives a bijective realisation with signed boundary map, which is also $m-$accretive. An existence criterion for boundary quadruples and boundary triples is established in terms of (V)-boundary conditions. The multiplicity of $M-$operators associated with a fixed (V)-boundary condition is addressed in an explicit way and a parametrisation of such operators is given. The theory is illustrated by a first-order ordinary differential operator and by the stationary diffusion equation, where $Q_0$ is identified as a Cayley transform of the Dirichlet-to-Neumann operator.