Boundary quadruples and bijective realisations of abstract Friedrichs operators

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 $$T_{\mathrm{\Theta }}\mathrm{i}\mathrm{s}\mathrm{b}\mathrm{i}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}⟺\text{lK}=\mathrm{\Theta }\dotplus \mathrm{\Gamma }(\ker T_1)\mathrm{.}$$ For bounded operator parameters $\phi :{\text{lK}}_1\to {\text{lK}}_0$ in the boundary space, we introduce the reference operator $$Q_0={\mathrm{\Gamma }}_1({\mathrm{\Gamma }}_0{\mathrm{\mid }}_{\ker T_1}{)}^{-1},$$ prove that $\mathrm{\parallel }{Q}_0\mathrm{\parallel }<1$, and obtain the exact criterion $$T_{\phi }\mathrm{i}\mathrm{s}\mathrm{b}\mathrm{i}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}⟺{\text{I}}_{{\text{lK}}_0}-\phi Q_0\mathrm{i}\mathrm{s}\mathrm{b}\mathrm{i}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{.}$$ 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.