Boundary relations and their Weyl families
Boundary relations and their Weyl families
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边界关系及其韦尔家族
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发表时间：
2006
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通讯作者：
H. Snoo
中科院分区：
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作者：
V. Derkach;S. Hassi;M. Malamud;H. Snoo
引入了边界关系和相应的Weyl族的概念。设S是闭对称线性算子，或者更一般地，是Hilbert空间h中的闭对称关系，H是辅助Hilbert空间，[图论]类似地定义了JH。Krein空间(h(2)，J(H))到Kre的酉关系G。在空间(H-2，J(H))中，称为伴随的S的边界关系，如果Ker Gamma=S，则对应的Weyl族M(λ)是de。作为帽(Lambda)上缺陷子空间(N)的象族，lambda是Gamma下CR的一个元素。在这里，Gamma不必是满足性的，甚至可以是多值的。虽然这导致了复Hilbert空间H上的某些全纯线性关系族与么正关系类Gamma：(H-2，J(H))->；(H-2，J(H))之间的丰富的联系，但它也推广了所谓的边值空间的概念，本质上扩展了抽象边界映射在边值问题联系中的适用性。此外，这些新概念给出了如下实现定理：每个H-值最大耗散(对lambda是C+的一个元素)全纯线性关系族是边界关系的Weyl族，如果满足某些极小条件，则该族在酉等价性下是唯一的。进一步研究了Weyl族的解析和谱理论性质与边界关系的几何性质之间的联系，并给出了一些应用。
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 h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C 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 H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(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 H-valued maximal dissipative (for lambda is an element of 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.