论文标题
有关光谱施瓦茨分布的研究
Studies on the Spectral Schwartz Distribution
论文作者
论文摘要
在复杂平面的开放子集上定义了算子在Banach空间中的分解函数，并且是全态。它遵守分解方程。定义了对Schwartz分布的该方程式的概括，并为该方程式所示的schwartz分布称为分解分布。在重要情况下，解析分布是整个平面。它的限制是连续的子集，是通常的分解函数。它的复杂共轭衍生物是光谱schwartz分布，它是由操作员的光谱集的子集携带的。光谱分布产生光谱分解。我们有广义的正交关系。完整性以自然方式定义，并且是这种情况，例如如果操作员有限。给出了矩阵和单一操作员的光谱分布。如果操作员是希尔伯特（Hilbert）空间上的自动伴侣操作员，则光谱分布是光谱家族的衍生物。我们计算具有非对称秩一个扰动的乘法运算符和乘法运算符的频谱分布和特征值问题。操作员不正常，可能具有离散的真实或虚构特征值或非平凡的约旦分解。
The resolvent function of an operator in a Banach space is defined on an open subset of the complex plane and is holomorphic. It obeys the resolvent equation. A generalization of this equation to Schwartz distributions is defined and a Schwartz distribution, which satisfies that equation is called a resolvent distribution. In important cases the resolvent distribution is the whole plane. Its restriction to the subset, where it is continuous, is the usual resolvent function. Its complex conjugate derivative is ,but a factor, the spectral Schwartz distribution, which is carried by a subset of the spectral set of the operator. The spectral distribution yields a spectral decomposition. We have a generalized orthogonality relation. Completeness is defined in a natural way and is the case e.g. if the operator is bounded. The spectral distribution of a matrix and a unitary operator are given. If the the operator is a self-adjoint operator on a Hilbert space, the spectral distribution is the derivative of the spectral family. We calculate the spectral distribution and the eigen value problem of the multiplication operator and of the multiplication operator with a non symmetric rank one perturbation. The operator is not normal and may have discrete real or imaginary eigenvalues or a nontrivial Jordan decomposition.
