On a functional calculus for decomposable operators and applications to normal, operator-valued functions
On a functional calculus for decomposable operators and applications to normal, operator-valued functions
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可分解运算符的泛函计算及其在普通运算符值函数中的应用
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发表时间：
1973
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通讯作者：
F. Gilfeather
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作者：
F. Gilfeather
当A = j时，©A()p(d)是直接积分上的可分解算子// = f。Hilbert空间的©H()p(d)和/是o-（A)邻域上的解析函数，则我们得到f{A()）几乎处处都有定义，f(A)(X)几乎处处都= f（A()）。利用此关系研究了可分离希尔伯特空间上的算子A，其中解析函数A是正规算子。得到了两个主要结果。设/是算子a谱的邻域上的解析函数。如果/ (z)对a谱中的所有z都是0，如果f(a)是正规算子，则a类似于二正规算子。已知二正规算子酉等价于一个正规算子与一个交换正规算子的2 × 2矩阵的直和。如上所述，如果f(A)是正规的，并且/(z) -£n对于n的谱中的每个T最多有两个根计数到它们的多重，则A是一个二元算子。所有空间都是复可分希尔伯特空间，其上的算子都是有界的。许多作者研究了当A的特定函数是正规函数时A的结构。A. Brown(3)研究了双正规算子，其中包括满足A = XI的算子。他给出了二正规算子的结构定理；最近在(2)和（15）中对这些结果进行了研究。J. Stampfli证明，如果A是正规的，A是可逆的，则A类似于正规算子；随后，S. Foguel和C. Apostol极大地推广了这一结果((1),(lO),（12),(17)）。在本文中，我们推广了这些结果，并证明了用算子约简理论来解决这些问题和类似的问题是可能的。关于算子结构的问题，如上面所研究的，将被简化为关于代数算子的问题，然后可以用更初级的算子理论方法来处理。这种方法似乎是新的，从本质上不同于以往的作品。
Whenever A = J. © A()p(d) is a decomposable operator on a direct integral // = f. © H()p(d) of Hilbert spaces and / is a function analytic on a neighborhood of o-(A), then we obtain that f{A()) is defined almost every- where and f(A)(X) = f(A()) almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic func- tion A is a normal operator. Two main results are obtained. Let / be an ana- lytic function on a neighborhood of the spectrum of an operator A. If / (z) 4 0 for all z in the spectrum of A and if f(A) is a normal operator, then A is similar to a binormal operator. It is known that a binormal operator is unitarily equivalent to the direct sum of a normal and a two by two matrix of commuting normal opera- tors. As above if f(A) is normal and in addition, /(z) — £n has at most two roots counted to their multiplicity for each T in the spectrum of N, then A is a binor- mal operator. In this paper, all spaces are complex separable Hilbert spaces and all opera- tors on them are bounded. Various authors have investigated the structure of A whenever a particular function of A is normal. A. Brown (3) studied binormal oper- ators which included operators satisfying A = XI. He gave a structure theorem for binormal operators; these results have more recently been investigated in (2), and (15). J. Stampfli has shown that if A" is normal and A is invertible, then A is similar to a normal operator; subsequently, S. Foguel and C. Apostol have considerably generalized this result ((l), (lO), (12), (17)). In this paper, we generalize these results and show that it is possible to approach these and simi- lar problems through reduction theory of operators. Questions, concerning the struc- ture of operators, such as those investigated above will be reduced to questions about algebraic operators, which can then be handled with more elementary opera- tor theory methods. This approach seems new and essentially different from pre- vious works.