Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $\Phi$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $\Phi$ is a unitary operator. As corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from Hto H +. Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case).
翻译:鉴于一个高度界定的对称操作员对Hilbert 空间V的A 0表示高度界定的对称操作员,我们将所有m- disistical exterations 在收缩 $\ Phi$: H-$\rightroll$ H + + 中,H Hand H + 是与边界四重相联的Hilbert 空间。如果而且只有在$\Phi$是单一操作员的情况下,这种扩展才会产生一个单一的C0组。作为从H到H +的单一操作员对称操作员的所有自对称扩展的对称。我们的结果扩展了冯纽曼启动的、由V. I. 和M. L. Gorbachuk、J. Behrndt 和M. Langer、S. A. Wegner 和其他许多人开发的边界三重理论,即边界始终存在四重线(即使对称的缺陷指数不同)。