Abstract

In this paper, certain spectral properties related with the first order linear differential-operator expression with involution in the Hilbert space of vector-functions at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential-operator expression with involution in the Hilbert spaces of vector-functions has been described. Then, the deficiency indices of the minimal operator have been calculated. Moreover, the space of boundary values of the minimal operator have been constructed. Afterwards, by using the method of Calkin–Gorbachuk, the general form of all selfadjoint extensions of the minimal operator in terms of boundary values has been found. Later on, the structure of spectrum of these extensions has been investigated.

中文翻译：

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带对合的一阶自伴微分算子

摘要

在本文中，研究了与向量函数的希尔伯特空间中有限距离对合的一阶线性微分算子表达式有关的某些谱性质。首先，描述了由一阶线性微分算子表达式在向量函数的希尔伯特空间中对合而生成的最小和最大算子。然后，计算了最小算子的缺陷指数。而且，已经构造了最小算子的边界值的空间。之后，通过使用Calkin–Gorbachuk方法，发现了极小算子在边界值方面的所有自伴扩展的一般形式。后来，对这些扩展的光谱结构进行了研究。