WEYL'S THEOREM FOR ALGEBRAICALLY k-QUASICLASS A OPERATORS
WEYL'S THEOREM FOR ALGEBRAICALLY k-QUASICLASS A OPERATORS
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DOI：
10.7494/opmath.2012.32.1.125
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发表时间：
2012
影响因子：
1
通讯作者：
Fugen Gao;X. Fang
中科院分区：
文献类型：
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作者：
Fugen Gao;X. Fang
如果 T 或 T 是作用于无限维可分希尔伯特空间的代数 k 拟类 A 算子，F 是与 T 交换的算子，并且存在一个正整数 n 使得 F n 具有有限秩，那么我们证明对于每个 f2 H( (T))，Weyl 定理对于 f(T)+F 成立，其中 H( (T)) 表示 (T) 邻域中所有解析函数的集合。此外，如果 T 是代数 k-拟类 A 算子，则 -Weyl 定理对于 f(T) 成立。此外，我们还证明，如果 T 或 T 是代数 k-拟类 A 算子，则对于每个 f2 H( (T))，T 的 Weyl 谱和近似点谱都遵循谱映射定理。我们从 Fredholm 理论的一些标准符号开始。在本文中，让 H 是一个具有内积的可分离复数希尔伯特空间；i 。设B(H)和K(H)分别表示所有有界线性算子的C代数和作用于H的紧算算子的理想。如果T 2 B(H)，我们将分别将T的零空间和范围写为kerT和ranT。还令 (T ) =dim kerT、(T ) =dim kerT 并令 (T )、a(T ) 分别表示 T 的频谱、近似点频谱。令 p = p(T ) 为 T 的上升度；即，满足 kerT p = kerT p+1 的最小非负整数 p 。如果这样的整数不存在，我们设 p(T ) =1。类似地，令 q = q(T ) 为 T 的下降；即，满足 ran T q =ranT q+1 的最小非负整数 q ，如果这样的整数不存在，则我们设置 q(T ) =1。众所周知，如果 p(T ) 和 q(T ) 都是有限的，则 p(T ) = q(T )。此外，当 是 T 解方程的极点时，0 < p( T ) = q( T ) < 1，请参见 Heuser（20，命题 50.2）。如果 ranT 为，则算子 T 2 B(H) 称为 Fredholm
If T or T is an algebraically k-quasiclass A operator acting on an infinite di- mensional separable Hilbert space and F is an operator commuting with T, and there exists a positive integer n such that F n has a finite rank, then we prove that Weyl's theorem holds for f(T)+F for every f2 H( (T)), where H( (T)) denotes the set of all analytic functions in a neighborhood of (T). Moreover, if T is an algebraically k-quasiclass A operator, then -Weyl's theorem holds for f(T). Also, we prove that if T or T is an algebraically k-quasiclass A operator then both the Weyl spectrum and the approximate point spectrum of T obey the spectral mapping theorem for every f2 H( (T)). We begin with some standard notation on Fredholm theory. Throughout this paper letH be a separable complex Hilbert space with inner producth ;i . Let B(H) and K(H) denote respectively, theC -algebra of all bounded linear operators and the ideal of compact operators acting on H. If T 2 B(H), we shall write kerT and ranT for the null space and the range of T respectively. Also let (T ) =dim kerT, (T ) =dim kerT and let (T ), a(T ) denote the spectrum, approximate point spectrum of T, respectively. Let p = p(T ) be the ascent of T; i.e., the smallest nonnegative integer p such that kerT p = kerT p+1 . If such an integer does not exist, we put p(T ) =1. Analogously, let q = q(T ) be the descent of T; i.e., the smallest nonnegative integer q such that ran T q =ranT q+1 , and if such an integer does not exist, we put q(T ) =1. It is well known that if p(T ) and q(T ) are both finite then p(T ) = q(T ). Moreover, 0 < p( T ) = q( T ) <1 precisely when is a pole of the resolvent of T, see Heuser (20, Proposition 50.2). An operator T 2 B(H) is called Fredholm if ranT is