Bishop's property (β) for paranormal operators
Bishop's property (β) for paranormal operators
复制标题
超自然算子的 Bishop 属性 (β)
DOI：
10.7153/oam-03-29
复制
发表时间：
2009
影响因子：
0.5
通讯作者：
K. Tanahashi
中科院分区：
文献类型：
--
作者：
A. Uchiyama;K. Tanahashi
对于可分复希尔伯特空间H上的算子T，我们说T具有毕晓普性质（β），如果对任何开子集D <$C和任何解析函数列fn：D →H，如n→∞一致地在每个紧子集K <$D上，则fn → 0一致地在K上。它是谱理论中一个非常重要的性质。每个正规算子（T <$T =TT <$）都具有Bishop性质（β）。现在，许多数学家试图将这个结果推广到非正规算子。本文证明了每一个超正规算子（对所有x ∈ H，∈ T 2 x <$$> x <$$> Tx <$2）都具有Bishop性质（β）.数学学科分类（2000）：47 B20。
For an operator T on a separable complex Hilbert space H , we say that T has Bishop’s property (β) if for any open subset D ⊂ C and any sequence of analytic functions fn : D →H such as ‖(T −z) fn(z)‖→ 0 as n→∞ uniformly on every compact subset K ⊂D , then fn → 0 uniformly on K . It is a very important property in spectral theory. It is well-known that every normal operator (T ∗T =TT ∗ ) has Bishop’s property (β ). Now, many mathematicians attempt to extend this result to non-normal operators. In this paper, we shall show that every paranormal operator (‖T 2x‖‖x‖ ‖Tx‖2 for all x ∈ H ) has Bishop’s property (β) . Mathematics subject classification (2000): 47B20.