Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let $S$ be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space $\mathfrak {H}$, let $\mathcal {H}$ be an auxiliary Hilbert space, let \[ J_\mathfrak {H}=\begin {pmatrix}0&-iI_\mathfrak {H} iI_\mathfrak {H} & 0\end {pmatrix}, \] and let $J_\mathcal {H}$ be defined analogously. A unitary relation $\Gamma$ from the KreÄ­n space $(\mathfrak {H}^2,J_\mathfrak {H})$ to the KreÄ­n space $(\mathcal {H}^2,J_\mathcal {H})$ is called a boundary relation for the adjoint $S^*$ if $\ker \Gamma =S$. The corresponding Weyl family $M(\lambda )$ is defined as the family of images of the defect subspaces $\widehat {\mathfrak {N}}_\lambda$, $\lambda \in \mathbb {C}\setminus \mathbb {R}$, under $\Gamma$. Here $\Gamma$ need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space $\mathcal {H}$ and the class of unitary relations $\Gamma :(\mathfrak {H}^2,J_\mathfrak {H})\to (\mathcal {H}^2,J_\mathcal {H})$, it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every $\mathcal {H}$-valued maximal dissipative (for $\lambda \in \mathbb {C}_+$) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.