关于Livsic的一个定理
On a theorem of Livsic
摘要：
对称的非自伴算子理论在解析函数的某些再生核希尔伯特空间的复函数理论中有一些深入的应用，以及在数学物理中对普通微分算子如薛定谔算子的研究中也有一些应用。简单对称算子的例子包括各种解析函数空间上的乘法算子，例如Hardy空间、deBranges-Rovnyak空间和Herglotz空间的模型子空间，普通微分算子（包括量子力学中的薛定谔算子），Toeplitz算子和无限Jacobi矩阵。本文发展了具有相等亏指数的简单对称算子的一般表示论，得到了一系列结果，改进和推广了Krein和Livsic的经典工作.特别是，我们提供了一个替代的证明定理的Livsic的特点时，两个简单的对称算子具有相等的亏损指数是酉等价的，我们提供了一个新的，更容易计算公式的Livsic特征函数的一个简单的对称算子具有相等的亏指数。
英文摘要：
The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operators, and infinite Jacobi matrices. In this paper we develop a general representation theory of simple symmetric operators with equal deficiency indices, and obtain a collection of results which refine and extend classical works of Krein and Livsic. In particular we provide an alternative proof of a theorem of Livsic which characterizes when two simple symmetric operators with equal deficiency indices are unitarily equivalent, and we provide a new, more easily computable formula for the Livsic characteristic function of a simple symmetric operator with equal deficiency indices.
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