CFI算子与(ω)性质和(ω1)性质的判定

殷乐 曹小红

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CFI算子与(ω)性质和(ω1)性质的判定

    作者简介: 殷乐(1994−),女,陕西人,硕士生,主要从事算子理论方面的研究. E-mail:yinle@snnu.edu.cn;
    通讯作者: 曹小红, xiaohongcao@snnu.edu.cn
  • 中图分类号: O177.2

CFI operators and the judgements of property (ω) and property (ω1)

    Corresponding author: CAO Xiao-hong, xiaohongcao@snnu.edu.cn
  • CLC number: O177.2

  • 摘要: 利用一致Fredholm指标性质定义了一个新的谱集,根据这个谱集给岀了算子T及其共轭算子T*满足$\left( {{\omega _1}} \right)$性质和$\left( \omega \right)$性质的判定条件.并且,对Hilbert空间上有界线性算子的$\left( \omega \right)$性质的与T交换的有限秩摄动F进行了讨论.
  • [1] Weyl H. Über beschränkte quadratische Formen, deren Differenz vollstetig ist[J]. Rendiconti Del Circolo Matematico Di Palermo, 1909, 27(1): 373-392. DOI:  10.1007/BF03019655.
    [2] 江樵芬, 钟怀杰. Banach代数直和上的乘子及其谱理论[J]. 系 统科学与数学, 2010, 30(1): 137-144. Jiang Q F, Zhong H J. Multipliers of the direct sum of banach algebras and the spectral theory[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30(1): 137-144.
    [3] 何建锋, 李耀堂. 矩 阵Hadamard积和Fan积的特征值界的改进[J]. 云南大学学报: 自然科学版, 2018, 40(1): 22-28. DOI:  10.7540/j.ynu.20170394. He J F, Li Y T. A modification on eigenvalue’s bounds of the Hadamard product and Fan product of matrices[J]. Journal of Yunnan University: Natural Sciences Edition, 2018, 40(1): 22-28.
    [4] Harte R, Lee W Y. Another note on Weyl’s theorem[J]. Transactions of the American Mathematical Society, 1997, 5: 349.
    [5] Rakočevic V. On a class of operators[J]. Matematicki Vesnik, 1985, 37: 423-426.
    [6] Rakočevic V. Operators obeying a-Weyl’s theorem[J]. Revue Roumaine des Mathematiques Pures et Appliquees, 1989, 34: 915-919.
    [7] Aiena P. Fredholm and local spectral theory, with applications to multipliers[M]. Netherlands: Springer, 2004.
    [8] Berberian S K. The Weyl spectrum of an operator[J]. Indiana University Mathematics Journal, 1970, 20(6): 529-544. DOI:  10.1512/iumj.1971.20.20044.
    [9] Harte R. Fredholm, Weyl and Browder theory[J]. Proceedings of the Royal Irish Academy, 1985, 85A(2): 151-176.
    [10] Harte R. Invertibility and singularity for bounded linear operators[M]. New York: Dekker, 1988.
    [11] Taylor A E. Theorems on ascent, descent, nullity and defect of linear operators[J]. Mathematische Annalen, 1996, 163(1): 18-49.
    [12] Aiena P, Pena P. Variation on Weyl’s theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 324(1): 566-579. DOI:  10.1016/j.jmaa.2005.11.027.
    [13] Sun C H, Cao X H, Dai L. Property $\left( {{\omega _1}} \right)$ and Weyl type theorem[J]. Journal of Mathematical Analysis and Applications, 2010, 363(1): 1-6. DOI:  10.1016/j.jmaa.2009.07.045.
    [14] Cao X H. Weyl spectrum of the products of operators[J]. Journal of the Korean Mathematical Society, 2008, 45(3): 771-780. DOI:  10.4134/JKMS.2008.45.3.771.
    [15] Xin Q L, Cao X H. Perturbations for property $\left( \omega \right)$ [J]. Journal of Graduate University of Chinese Academy of Sciences, 2012, 29(5): 586-593.
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出版历程
  • 收稿日期:  2019-07-01
  • 录用日期:  2020-01-19
  • 网络出版日期:  2020-04-11

CFI算子与(ω)性质和(ω1)性质的判定

    作者简介:殷乐(1994−),女,陕西人,硕士生,主要从事算子理论方面的研究. E-mail:yinle@snnu.edu.cn
    通讯作者: 曹小红, xiaohongcao@snnu.edu.cn
  • 陕西师范大学 数学与信息科学学院,陕西 西安 710119

摘要: 利用一致Fredholm指标性质定义了一个新的谱集,根据这个谱集给岀了算子T及其共轭算子T*满足$\left( {{\omega _1}} \right)$性质和$\left( \omega \right)$性质的判定条件.并且,对Hilbert空间上有界线性算子的$\left( \omega \right)$性质的与T交换的有限秩摄动F进行了讨论.

English Abstract

  • 1909年,Weyl检查Hilbert空间上正规算子的谱时发现了Weyl定理[1]. 自此谱理论逐渐发展成为算子理论和算子代数中的一个重要分支,与解算子方程联系密切,且应用广泛[2-3]. 之后许多文献都讨论了Weyl定理的变式[4-6]. Rakočevic[5]研究了其中一个变式,即 $\left( \omega \right)$ 性质. 在本文中,H表示复的无限维可分的Hilbert空间,$B\left( H \right)$ 表示 $H$ 上有界线性算子的全体. 对于 $T \in B\left( H \right)$$n\left( T \right)$$d\left( T \right)$ 分别表示算子 $T$ 的零空间 $N\left( T \right)$ 的维数和值域 $R\left( T \right)$ 的余维数. 由文献[7]中推论1.15可知,若 $d\left( T \right) < 0$,则 $R\left( T \right)$ 闭. 并且,我们定义:

    $\qquad {\sigma _d}(T) = \left\{ {\lambda \in {\bf{C}}:R\left( {T - \lambda I} \right){\text{不闭}}} \right\}. $

    回顾可知,算子 $T \in B\left( H \right)$ 称为上半Fredholm算子,若 $T$ 的值域 $R\left( T \right)$ 闭且其零空间 $N\left( T \right)$ 是有限维的[8-10];算子 $T \in B\left( H \right)$ 称为下半Fredholm算子,若 $T$ 的值域 $R\left( T \right)$ 的余维数是有限维的. 特别地,若 $R\left( T \right)$ 闭且 $n\left( T \right) = 0$,则称算子 $T$ 为下有界算子;若 $d\left( T \right) = 0$,则称算子 $T$ 为满算子. 称算子 $T$ 是可逆的,当且仅当 $N\left( T \right) = d\left( T \right) = 0$. 设算子 $T$ 的谱 $\sigma \left( T \right)$,逼近点谱 ${\sigma _a}\left( T \right)$ 和满谱 ${\sigma _s}\left( T \right)$ 定义如下:

    $\qquad \sigma (T) = \left\{ {\lambda \in {\bf{C}}:T - \lambda I{\text{不为可逆算子}}} \right\}; $

    $\qquad {\sigma _a}(T) = \left\{ {\lambda \in {\bf{C}}:T - \lambda I{\text{不为下有界算子}}} \right\}; $

    $\qquad {\sigma _s}(T) = \{ \lambda \in {\bf{C}}:T - \lambda I{\text{不为满算子}}\}. $

    令相应的预解集为 $\rho (T) ={\bf{C}}\backslash \sigma (T)$${\rho _a}(T) = {\bf{C}}\backslash {\sigma _a}(T)$${\rho _s}(T) = {\bf{C}}\backslash {\sigma _s}(T)$. 若算子 $T$ 是半(上半或下半)Fredholm算子,则可定义算子 $T$ 的指标为 ${\rm{ind}}\left( T \right) = n\left( T \right) - d\left( T \right)$. 若 ${\rm{ind}}\left( T \right) < \infty $,则称算子 $T$ 为Fredholm算子. 特别地,当 ${\rm{ind}}\left( T \right) = 0$ 时,称 $T$ 为Weyl算子. 设算子 $T$ 的本质谱 ${\sigma _e}\left( T \right)$ 和Weyl谱 ${\sigma _w}\left( T \right)$ 定义如下:

    $\qquad {\sigma _e}(T) = \{ \lambda \in {\bf{C}}:T - \lambda I{\text{不为}}{\rm{ Fredholm }}{\text{算子}}\}; $

    $\qquad {\sigma _w}(T) = \{ \lambda \in {\bf{C}}:T - \lambda I{\text{不为}}{\rm{Weyl }}{\text{算子}}\}. $

    $T$ 是Fredholm算子且有有限的升标 ${\rm{asc}}\left( T \right)$ 和降标 ${\rm{des}}\left( T \right)$ 时,称 $T$ 是Browder算子,其中 ${\rm{asc}}\left( T \right) = \inf \left\{ {n \in {\bf{N}}:N\left( {{T^n}} \right) = N\left( {{T^{n + 1}}} \right)} \right\}$${\rm{des}}\left( T \right) = \inf \left\{ {n \in {\bf{N}}:R\left( {{T^n}} \right) = R\left( {{T^{n + 1}}} \right)} \right\}$. 由文献[11]中推论4.4可知Browder算子就是有有限升降标的Weyl算子. 我们定义Browder谱为

      ${\sigma _b}(T) = \{ \lambda \in {\bf{C}}:T - \lambda I{\text{不为}}{\rm{Browder}}{\text{算子}}\} $

    ${\;\rho _b}(T) = {\bf{C}}\backslash {\sigma _b}(T)$.给定一个子集 $G \subseteq {\bf{C}}$,令 $\operatorname{int} G$$G$ 的所有内点之集,${\rm{iso}}G$$G$ 的所有孤立点之集. 令 ${\rm{acc}}G = \bar G\backslash {\rm{iso}}G$,其中 $\bar G$$G$ 的闭包. 并且,称 $\partial G$$G$的边界. 称 $T \in B\left( H \right)$ 为isoloid算子,若 ${\rm{iso}}\sigma (T) \subseteq {\sigma _p}(T)$,其中 ${\sigma _p}(T) = $$ \{ \lambda \in {\bf{C}}:n\left( {T - \lambda I} \right) > 0\} $.

    在文献[12~13]中,作者介绍了 $\left( \omega \right)$ 性质和 $\left( {{\omega _1}} \right)$ 性质的相关概念. 称 $T$ 满足 $\left( \omega \right)$ 性质,若

    $\qquad{\sigma _a}(T)\backslash {\sigma _{ea}}(T) = {{\rm{\pi }}_{00}}(T),$

    其中 ${{\rm{\pi }}_{00}}(T) = \left\{ {\lambda \in {\rm{iso}}\sigma (T):0 < n(T - \lambda I) < \infty } \right\}$${\sigma _{ea}}(T) = {\bf{C}}\backslash {\rho _{ea}}(T)$${\rho _{ea}}(T) = \{ \lambda \in {\bf{C}}:T - \lambda I{\text{是上半}}{\rm{Fredholm}}$算子且 ${{\rm{ind}}( {T - \lambda I{\rm{ }}} ) \leqslant 0}\} $. 称 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质,若 ${\sigma _a}(T)\backslash {\sigma _{ea}}(T) \subseteq {{\rm{\pi }}_{00}}(T). $

    在本文中,我们用CFI算子[14]定义了一个新的谱集,利用这个谱集讨论了算子 $T$ 及其共轭算子 ${T^ * }$ 满足 $\left( \omega \right)$ 性质的判定条件,并且讨论了算子 $T$ 关于 $\left( \omega \right)$ 性质的有限秩摄动.

    在文献[14]中,作者定义了一致Fredholm指标算子(简写为CFI算子),即算子 $T \in B\left( H \right)$ 称为CFI算子,若对任意的 $S \in B\left( H \right)$,下列情况之一成立:

    (1)$TS$$ST$ 均为Fredholm算子且 ${\rm{ind}}\left( {TS} \right) = {\rm{ind}}\left( {ST} \right) = {\rm{ind}}\left( S \right)$

    (2)$TS$$ST$ 都不为Fredholm算子.

    并且,作者证明了 $T$ 是CFI算子当且仅当以下情况之一成立:

    (1)$T$ 是Weyl算子;

    (2)$R\left( T \right)$不闭;

    (3)$R\left( T \right)$ 闭且 $n\left( T \right) = d\left( T \right) = \infty $.

    显然,$T$ 不是CFI算子当且仅当T是半Fredholm算子且 ${\rm{ind}}\left( T \right) \ne 0$.

    利用CFI算子,我们定义了一个新的谱集:${\sigma _1}\left( T \right) = \left\{ {\lambda \in {\bf{C}}:T - \lambda I{\text{不为}}{\rm{CFI}}{\text{算子}}} \right\}$. 显然 ${\sigma _1}\left( T \right)$ 是一个开集. 令 ${\rho _1}(T) = {\bf{C}}\backslash {\sigma _1}(T)$. 接下来,我们利用 ${\sigma _1}\left( T \right)$ 分别研究算子 $T$ 及其共轭算子 ${T^ * }$$\left( {{\omega _1}} \right)$ 性质和 $\left( \omega \right)$ 性质.

    定理1 设 $T \in B\left( H \right)$. $T$ 满足 $\left( {{\omega _1}} \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    $\partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > d\left( T \right. - \left. {\left. {\lambda I} \right)} \right\} \cup \{ \lambda \in $${\mathop{\rm int}} {\rho _1}\left( T \right):n\left( {T - \lambda I} \right) = \infty \} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right) $.

    证明 显然,对任意 $T \in B\left( H \right)$,有    ${\sigma _b}\left( T \right) \supseteq \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in{\bf{C}}:n\left( {T \!-\! \lambda I} \right)} \right. \!>\! \left. {d\left( {T \!-\! \lambda I} \right)} \right\} \cup \left\{ \lambda \in \operatorname{int} {\rho _1}\left( T \right)\!:\right.$ $\left.n\left( {T \!-\! \lambda I} \right) = { \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right)$.

    对于反包含,我们假设     ${\lambda _0} \notin \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}} \right.\left( T \right):n\left( T \right. \left. { - \lambda I} \right)\left. { = \infty } \right\} \cup $$\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup{\sigma _d}\left( T \right) $,则有 $n\left( {T - {\lambda _0}I} \right) \leqslant d\left( {T - {\lambda _0}I} \right)$$R\left( {T - {\lambda _0}I} \right)$ 闭. 不妨设 ${\lambda _0} \in {\sigma _a}\left( T \right)$. 由于 ${\lambda _0} \notin \partial {\sigma _1}\left( T \right)$,故 ${\lambda _0} \in {\sigma _1}\left( T \right) \cup \operatorname{int} {\rho _1}\left( T \right)$. 若 ${\lambda _0} \in {\sigma _1}\left( T \right)$,则 ${\lambda _0} \in {\sigma _a}(T)\backslash {\sigma _{ea}}(T)$,由 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质可知 ${\lambda _0} \in {\rho _b}\left( T \right)$,这与 ${\lambda _0} \in {\sigma _1}\left( T \right)$ 矛盾. 若 ${\lambda _0} \in \operatorname{int} {\rho _1}\left( T \right)$,则 $n\left( {T - {\lambda _0}I} \right) < \infty $. 因此 ${\lambda _0} \in {\rho _w}\left( T \right)$,即 ${\lambda _0} \in {\sigma _a}(T)\backslash {\sigma _{ea}}(T)$. 同理,由 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质可知 ${\lambda _0} \in {\rho _b}\left( T \right)$.

    反之,假设 ${\lambda _0} \in {\sigma _a}(T)\backslash {\sigma _{ea}}(T)$. 由半Fredholm算子的摄动定理可知 ${\lambda _0} \in {\sigma _1}\left( T \right) \cup \operatorname{int} {\rho _1}\left( T \right)$,故 ${\lambda _0} \notin \partial {\sigma _1}\left( T \right)$. 因此     ${\lambda _0} \notin \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}} \right.\left( T \right):n\left( T \right.\left. { - \lambda I} \right) = \left. \infty \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup$$ {\sigma _d}\left( T \right)$. 故 ${\lambda _0} \notin {\sigma _b}\left( T \right)$,即 ${\lambda _0} \in {{\rm{\pi }}_{00}}(T)$. 证毕.

    由定理1可得,下列叙述等价:

    (1) $T$ 满足 $\left( {{\omega _1}} \right)$ 性质;

    (2) $\sigma \left( T \right) \subseteq {\rho _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]$${\sigma _w}\left( T \right) = {\sigma _b}\left( T \right)$

    (3) $\sigma \left( T \right) = \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):n\left( T \right.\left. { - \lambda I} \right) = \infty } \right\} \cup \left. {\left[ {{\rho _a}\left( T \right)} \right. \cap \sigma \left( T \right)} \right] \cup $       $ {\sigma _d}\left( T \right) \cup{\sigma _0}\left( T \right) $

    其中 ${\sigma _0}\left( T \right) = \sigma \left( T \right)\backslash {\sigma _b}\left( T \right)$.

    类似定理1的证明,我们可以得到 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质的充要条件.

    推论1 设 $T \in B\left( H \right)$. ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) =$    $ \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. < d\left( T \right. - \left. {\left. {\lambda I} \right)} \right\} \cup \{ \lambda \in {\mathop{\rm int}} {\rho _1}\left( T \right):d\left( {T - \lambda I} \right) = \infty \} \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$.

    根据之前的结果,我们很自然地考虑了 $T$${T^ * }$ 均满足 $\left( {{\omega _1}} \right)$ 性质时 ${\sigma _b}\left( T \right)$ 的结构.

    推论2 设 $T \in B\left( H \right)$. 则下列命题成立:

    (1)$T$${T^ * }$ 均满足 $\left( {{\omega _1}} \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    $ \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):d\left( {T - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}} \right.\left( T \right)\left. { \cap \sigma \left( T \right)} \right] \cup\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$

    (2)若 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质. 则 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    $ \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:d\left( {T - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$

    (3)若 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质. 则 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    $\partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:d\left( {{T^ * } - \bar \lambda I} \right) = \infty } \right\}\cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$

    (4)若 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质. 则 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    $\partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:d\left( {{T^ * } - \bar \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right)$.

    证明 (1)根据定理1和推论1,可得    ${\sigma _b}( T ) = \{ \partial {\sigma _1}( T ) \cup \{ \lambda \in {\bf{C}}:n( {T - \lambda I} ) > d( {T - \lambda I} ) \} \cup \{ \lambda \in \operatorname{int} {\rho _1}( T ):$ $n( {T - \lambda I} ) = \infty \} \cup [ {\rho _a}( T ) \cap \sigma ( T ) ] \cup $    ${\sigma _d}( T )\}\cap \{ \partial {\sigma _1}( T ) \cup\{ \lambda \in {\bf{C}}:n( {T - \lambda I} ) < d {( T { - \lambda I} )} \} \cup \{ \lambda \in \operatorname{int} {\rho _1}( T ):d( {T - \lambda I} ) = \infty \} \cup [ {\rho _s}( T ) \cap\sigma ( T ) ] \} = $    $ \partial {\sigma _1}( T ) \cup $$ \{ \lambda \in \partial {\sigma _1}( T ):n( {T - \lambda I} ) < d( {T - \lambda I} ) \} \cup[ {{\rho _s}( T ) \cap \sigma ( T )} ] \cup $    $ \{ \lambda \in {\mathop{\rm int}} {\rho _1}( T ):R( {T - \lambda I} ){\text{不闭且}}n( {T - \lambda I} ) =d( {T - \lambda I} ) = \infty \} \cup $    $\{ {\lambda \in {\mathop{\rm int}} {\rho _1}( T ):R( {T - \lambda I} ){\text{闭且}}n( {T \!-\! \lambda I} ) \!=\! d( {T \!-\! \lambda I} ) \!=\! \infty } \} \cup \{ {\lambda \in {\mathop{\rm int}} {\rho _1}( T ):R( {T \!-\! \lambda I} ){\text{不闭且}}} n({T \!-\! \lambda I} ){\rm{ = }}\infty \} \cup$    $ [ {{\rho _a}( T ) \cap \sigma ( T )} ] \cup [ {\partial {\sigma _1}( T ) \cap {\sigma _d}( T )} ] \cup \{ {\lambda \in {\sigma _d}( T ):n( {T - \lambda I} ) < d( {T - \lambda I} )} \} \cup \left\{ \lambda \in {\mathop{\rm int}} {\rho _1}( T ):R( {T - \lambda I} )\right. $不闭且    $\left.d( {T - \lambda I} ) = \infty \right\} \subseteq $    $ \partial {\sigma _1}( T ) \cup\{ {\lambda \in {\mathop{\rm int}} {\rho _1}( T ):d( {T - \lambda I} ) = \infty } \} \cup [ {{\rho _a}} ( T ) { \cap \sigma ( T )} ] \cup [ {{\rho _s}( T ) \cap \sigma ( T )} ] \cup {\sigma _d}( T ){\rm{ = }}$    $ \partial {\sigma _1}( T ) \cup \{ {\lambda \in {\mathop{\rm int}} {\rho _1}( T ):d( {T - \lambda I} ) = \infty } \} \cup [ {{\rho _a}} ( T ) \cap\sigma ( T ) ] \cup [ {\rho _s}( T ) \cap \sigma ( T ) ] \cup $    $[ \partial {\sigma _1}( T ) \cap {\sigma _d}( T ) ] \cup [ \operatorname{int} {\rho _1}( T ) \cap {\sigma _d}( T ) ] = $    $ \partial {\sigma _1}( T ) \cup \{ \lambda \in \operatorname{int} {\rho _1}( T ): d( T - {\lambda I} ) = \infty \} \cup [ {\rho _a}( T ) \cap \sigma ( T ) ] \cup [ {{\rho _s}( T ) \cap \sigma ( T )} ] \subseteq {\sigma _b}( T )$. 因此可得

    $\qquad{\sigma _b}\left( T \right) = \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):d\left( {T - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right].$

    反之,由所给条件可得     ${\sigma _b}\left( T \right) = \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):d\left( {T - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]\cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq \partial {\sigma _1}\left( T \right) \cup$        $ \left\{ {\lambda \in {\mathop{\rm int}} {\rho _1}\left( T \right):R\left( {T - \lambda I} \right){\text{闭且}}n\left( {T - \lambda I} \right) = d\left( {T - \lambda I} \right) = \infty } \right\} \cup$        $ \left\{ {\lambda \in } {\mathop{\rm int}} {\rho _1}\left( T \right):R\left( {T - \lambda I} \right){\text{不闭且}}n\left( {T - \lambda I} \right) = d\left( {T - \lambda I} \right) =\right.\left. \infty \right\} \cup $        $\left\{ {\lambda \in {\mathop{\rm int}} {\rho _1}\left( T \right):R\left( {T - \lambda I} \right){\text{不闭且}}} \right.n ( T { - \lambda I}) < d\left( {T - \lambda I} \right) = \infty \} \cup $        $\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \{ \lambda \in {{\bf{C}}}:n\left( {T - \lambda I} \right) > d\left( {T - \lambda I} \right)\} \subseteq $        $\partial {\sigma _1}\left( T \right) \cup\{ \lambda \in{{\bf{C}}}:$$n\left( {T \!-\! \lambda I} \right)\left. { > {d\left( {T\! -\! \lambda I} \right)} } \right\} \cup \left\{ {\lambda \in {\mathop{\rm int}} {\rho _1}\left( T \right):n\left( {T \!-\! \lambda I} \right) \!=\! \infty } \right\} \cup $        $\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right) \subseteq {\sigma _b}\left( T \right) $. 因此,由定理1可得 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质. 同样地,由推论1我们可以证明 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质.

    (2)根据推论2(1)的必要性,可得     ${\sigma _b}\left( T \right) = \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):d\left( {T - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}} \left( T \right)\cap \sigma \left( T \right) \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq$    $ \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:d\left( {T - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}} \right.\left( T \right)\left. { \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap } \right.$$\left. {\sigma \left( T \right)} \right]$,且其反包含显然成立.

    反之,因为 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质,则由定理1可得     ${\sigma _b}\left( T \right) \!=\! \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T \!-\! \lambda I} \right)} \right. > d\left( T \!-\! \left. {\lambda I}\right) \right\} \cup \{ \lambda \in$$ \operatorname{int} {\rho _1}\left( T \right):n\left( {T - \lambda I} \right) = \infty \} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right)$. 由条件得     ${\sigma _b}\left( T \right) = \left\{ {\partial {\sigma _1}\left( T \right)} \right. \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > d\left( T \right. - \left. {\left. {\lambda I} \right)} \right\} \cup\{ \lambda \in \operatorname{int} {\rho _1}( T ):n( {T - \lambda I}) = \infty \} \cup $        $ [ {{\rho _a}( T ) \cap \sigma ( T )} ] \cup {\sigma _d}( T ) \} \cap\{ {\partial {\sigma _1}( T ) \cup \{ {\lambda \in {\bf{C}}:d( {T - \lambda I}) = \infty } \}} \cup $        $ [ {{\rho _a}( T ) \cap \sigma ( T )} ] \cup[ {\rho _s}( T ) \cap\sigma ( T ) ] \} = $        $\partial {\sigma _1}( T ) \cup $        $\{ {\lambda \in } \partial {\sigma _1}( T ):d\left( T \right. - \left. {\left. {\lambda I} \right) = \infty } \right\} \cup $        $\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {\partial {\sigma _1}\left( T \right) \cap {\sigma _d}\left( T \right)} \right] \cup $        $\left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):n\left( {T\!\! -\! \!\lambda I} \right) \!=\! d\left( {T \!\!-\!\! \lambda I} \right)} \right.\!=\! \infty \} \cup \{ \lambda \in {\sigma _d}( T ):d( T \!-\! {\lambda I}) \!=\! \infty \} \cup $        $ \left\{ {\lambda \in {\rho _a}\left( T \right) \cap \sigma \left( T \right):d\left( T \right. \!\!-\!\! \left. {\left. {\lambda I} \right) \!=\! \infty } \right\}} \right. \cup\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]\subseteq $        $ \partial {\sigma _1}\left( T \right) \cup \{ \lambda \in {\bf{C}}:n( {T - \lambda I} ) < d( T - {\lambda I} ) \} \cup \left\{ {\lambda \in \operatorname{int} {\rho _1}\left( T \right):d\left( {T - \lambda I} \right) = \infty } \right\} \cup $        $\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq $${\sigma _b}\left( T \right)$. 因此,由推论1可得 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质.

    类似于推论2(2)的证明,易得推论2(3)(4)成立. 证毕.

    接下来,我们讨论 $T$ 满足 $\left( \omega \right)$ 性质时的相关结论.

    定理2 若 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质. 则 $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    ${\left[ \rho \right._1}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\} \cup \left. {\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]$.

    证明 由推论2(3)可知,当 $T$${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质时,有    ${\sigma _b}\left( T \right) = \partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {{\bf{C}}}:} \right.d\left( {{T^ * } - \bar \lambda I} \right)\left. { = \infty } \right\} \cup$$ \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$. 因为     $\partial {\sigma _1}\left( T \right) \!=\!\! \left\{ {\lambda \in\! \partial {\rho _1}\left( T \right) \cap {\rm{iso}}\sigma \left( T \right):\!} \right.\!n\left. {\left( {T \!\!-\!\! \lambda I} \right) \!=\! \infty } \right\}\cup \{ \lambda \in\! \partial {\rho _1}\left( T \right) \cap {\rm{iso}}\sigma ( T ): $ $ n ( {T \!\!-\! \lambda I} ){\rm{<}}\infty \} \!\cup\! \left[ {\partial {\rho _1}\left( T \right) \!\cap \!{\rm{acc}}\sigma \left( T \right)} \right] \!\subseteq $        $\left\{ {\lambda \!\in\! {\bf{C}}:} \right.n\left. {\left( {T \!-\! \lambda I} \right) \!= \!\infty } \right\} \!\cup\! \left\{ {\lambda \!\in } \right.\partial {\rho _1}\left( T \right) \cap {\rm{iso}}\sigma \left( T \right):n\left. {\left( {T - \lambda I} \right){\rm{ < }}\infty } \right\} \cup \left[ {{\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right]$,并且有     $\left\{ {\lambda \!\in\! {\bf{C}}:d\left( {{T^ * } \!-\! \bar \lambda I} \right) \!=\! \infty } \right\} \!=\! \left\{ {\lambda \notin } \right.\left. {{\sigma _d}\left( T \right):n\left( {T - \lambda I} \right) \!=\! \infty } \right\} \cup \left\{ {\lambda \in {\sigma _d}\left( T \right):d\left( {{T^ * } - \bar \lambda I} \right) = \infty } \right\} \subseteq$    $ \left\{ {\lambda \notin } {\sigma _d}\left( T \right):n\left( {T - \lambda I} \right) =\right.\left. { \infty } \right\} \cup \left[ {{\rho _1}\left( T \right)} \right.\left. { \cap \sigma \left( T \right)} \right] \subseteq \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right) = \infty } \right\} \cup$    $ \left[ {{\rho _1}\left( T \right) \cap } \right.{\rm{acc}}\left. {\sigma \left( T \right)} \right] \cup \left[ {{\rho _1}\left( T \right) \cap } \right.{\rm{iso}}\left. {\sigma \left( T \right)} \right] \subseteq $    $\left\{ {\lambda \in {\bf{C}}:n} \right.\left( T \right. -\left. {\left. {\lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _1}\left( T \right) \cap } \right.{\rm{acc}}\left. {\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\rho _1}\left( T \right) \cap {\rm{iso}}\sigma \left( T \right):} \right.n\left. {\left( {T - \lambda I} \right) < \infty } \right\} \cup $    $\left\{ {\lambda \in {\rho _1}\left( T \right) \cap {\rm{iso}}\sigma \left( T \right):} \right.n\left. {\left( {T - \lambda I} \right) = \infty } \right\} \subseteq $    $\{ \lambda \in {\bf{C}}:n( {T - \lambda I} ) = \infty \} \cup \left[ {{\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\rho _1}\left( T \right) \cap {\rm{iso}}} \right.\sigma \left( T \right):n\left( T \right. - $$\left. {\left. {\lambda I} \right)< \infty } \right\}.$

    $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子,则有 $\left\{ {\lambda \in \partial {\rho _1}\left( T \right) \cap {\rm{iso}}\sigma \left( T \right):} \right.n\left. {\left( {T - \lambda I} \right){\rm{ < }}\infty } \right\} \cap {\sigma _b}\left( T \right) = \emptyset$,且 $ \left\{ {\lambda \in {\rho _1}\left( T \right) \cap} \right.$$\left. {\rm{iso}}\sigma \left( T \right):n {\left( {T \!-\! \lambda I} \right){\rm{ < }}\infty } \right\} \!\cap \!{\sigma _b}\left( T \right) \!=\! \emptyset $. 故有     ${\sigma _b}\left( T \right) \!=\! \left\{ {\partial {\sigma _1}\left( T \right)\! \cup\! } \right.\left\{ {\lambda \!\in\! {\bf{C}}:} \right.d\left( {{T^ * }} \right. \!-\! \bar \lambda \left. {\left. I \right)\left. { \!=\! \infty } \right\} \!\cup \!\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \!\cup\! \left[ {{\rho _s}\left( T \right) \!\cap\! \sigma \left( T \right)} \right]} \right\} \!\cap {\sigma _b}\left( T \right) = $        $\left[ {{\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n} \right.\left( T \right. -\left. {\lambda I} \right) = \left. \infty \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq $        $\left[ {{\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \{ \lambda \in {\bf{C}}:n( {T - \lambda I} ) > d( T $$\left. {\left. { - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {{\bf{C}}}:n\left( {T - \lambda I} \right){\rm{ = }}\infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq $         ${\sigma _b}\left( T \right). $因此     ${\sigma _b}\left( T \right) = \left[ {{\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \{ \lambda \in {{\bf{C}}}: $$n( {T - \lambda I} ) > d( {T - \lambda I} ) \} \cup \{ \lambda \in {{\bf{C}}}:n( T - \lambda I) = \infty \} \cup [ {\rho _a}( T ) \cap \sigma ( T ) ]$.

    反之,由于有    ${\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right) = \left[ {\partial {\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left[ {\operatorname{int} {\rho _1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \subseteq $    $\partial {\sigma _1}\left( T \right) \cup\left[ {{\rho _w}} \right.\left. {\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left[ {{\sigma _d}\left( T \right) \!\cap \!{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \!\in\! {\rm{acc}}\sigma \left( T \right)\backslash {\sigma _d}\left( T \right):n\left( {T \!-\! \lambda I} \right) \!=\! d\left( {T \!-\! \lambda I} \right) \!=\! \infty } \right\}\subseteq $    $ \partial {\sigma _1}\left( T \right) \cup \left[ {{\rho _w}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:} \right.$$d\left( {{T^ * } \!-\! \bar \lambda I} \right)\left. { \!=\! \infty } \right\} $,    $\left\{ {\lambda \!\in\! {\bf{C}}\!:\!n\left( {T - \lambda I} \right) \!=\! \infty } \right\} \!=\! \left\{ \lambda \right.\left. { \notin {\sigma _d}\left( T \right)\!:\!d\left( {{T^ * } \!-\! \bar \lambda I} \right) \!=\! \infty } \right\} \cup \left\{ {\lambda \!\in \!{\sigma _d}\left( T \right)\!:\!} \right.n\left( {T \!-\! \lambda I} \right)\left. { \!=\! \infty } \right\} \subseteq $    $\left\{ {\lambda \in {\bf{C}}\!:\!} \right.d\left( {{T^ * } \!-\! \bar \lambda I} \right)=\left. { \infty } \right\} \cup {\sigma _d}\left( T \right)$,并且    $\left\{ {\lambda \in {\bf{C}}:} \right.n\left( {\left. {T \!-\! \lambda I} \right)} \right.\left. { > d\left( {T \!-\! \lambda I} \right)} \right\} \!=\! \left\{ {\lambda \in {\bf{C}}:n\left( {T \!-\! \lambda I} \right)} \right. > \left. {d\left( {T \!-\! \lambda I} \right) \!=\! 0} \right\} \cup \left\{ {\lambda \in {\bf{C}}} \right.:n\left( {T \!-\! \lambda I} \right) > d\left( T \right. \!-\!\left. {\left. {\lambda I} \right) > 0} \right\} = $    $ \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right) > 0} \right\}$. 由 ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质,可得     $\{ \lambda \in {\bf{C}}:n\left( T- { \lambda I} \right) > $$ d\left( {T - \lambda I} \right) > 0\} \cap {\sigma _b}\left( T \right) = \emptyset$ 且     $\left[ {{\rho _w}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cap {\sigma _b}\left( T \right) = \emptyset .$ 因此,由所给条件可得     ${\sigma _b}\left( T \right) = \{ {{[ \rho }_1}\left( T \right) \cap $${\rm{acc}}\sigma ( T ) ] \cup \{ \lambda \in {\bf{C}}:n( {T - \lambda I} ) > d( {T - \lambda I} ) \} \cup \{ {\lambda \in } {\bf{C}}: {n( {T - \lambda I} ) = \infty } \} \cup \left[ {{\rho _a}} \right.\left( T \right)\left. {\left. { \cap \sigma \left( T \right)} \right] } \right\} \cap $        ${\sigma _b}\left( T \right) =\partial {\sigma _1}\left( T \right) \cup \left\{ {\lambda \in {\bf{C}}:} \right. d( {T^ * } - \bar \lambda I )\left. { = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap } \right.$$\left. {\sigma \left( T \right)} \right] \cup {\sigma _d}\left( T \right)$. 则由推论2(4)可得 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质.

    并且,假设 ${\lambda _0} \in {{\rm{\pi }}_{00}}\left( T \right)$. 若 ${\lambda _0} \!\in\! \left\{ {\lambda \!\in\! {\bf{C}}\!:\!n\left( {T \!-\! \lambda I} \right)} \right. \! > \! \left. {d\left( {T \!-\! \lambda I} \right)} \right\}$,则 ${\lambda _0} \in {\rho _b}\left( T \right)$. 若 ${\lambda _0} \notin \left\{ {\lambda \in } \right.$${\bf{C}}:n\left( {T - \lambda I} \right) > \left. {d\left( {T - \lambda I} \right)} \right\}$,则 ${\lambda _0} \notin \left. {{{\left[ \rho \right.}_1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T \!-\! \lambda I} \right)} \right. > \left. {d\left( {T \!-\! \lambda I} \right)} \right\} \cup \left\{ \lambda \right. \in $${\bf{C}}:\left. {n\left( {T \!-\! \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]$,故 ${\lambda _0} \in {\rho _b}\left( T \right) \subseteq $${\;\rho _{ea}}\left( T \right) $. 因此 ${{\rm{\pi }}_{00}}\left( T \right) \subseteq {\sigma _a}\left( T \right)\backslash {\sigma _{ea}}\left( T \right)$.

    $\left\{ {\lambda \in {\rm{iso}}\sigma{ \left( T \right):n\left( {T \!-\! \lambda I} \right) = 0}} \right\} \cap \left\{ {\left. {{{\left[ \rho \right.}_1}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T \!-\! \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\}} \right. \cup \{ {\lambda \in } {\bf{C}}: n( {T \!-\! \lambda I} ) \!=\! \infty \} \cup $$[ {\rho _a}( T ) \cap \sigma ( T ) ]\} = \emptyset $,可得 ${\rm{iso}}\sigma \left( T \right) \subseteq {\sigma _p}\left( T \right)$. 这就证明了 $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子. 证毕.

    注1 (1)若 $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子,则有     ${\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.n\left( {T - \lambda I} \right) > $$\left. {d\left( {T - \lambda I} \right)} \right\} \cup \left. {\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]$.

    (2)定理2中条件“${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质”是本质的. 例如,设 ${T_1},{T_2} \in B\left( {{\ell ^2}} \right)$ 定义为:

    $\qquad{T_1}\left( {{x_1},{x_2},{x_3}, \cdots } \right) = \left( {0,{x_1},{x_2},{x_3}, \cdots } \right),$

    $\qquad{T_2}\left( {{x_1},{x_2},{x_3}, \cdots } \right) = \left( {{x_2},{x_3},{x_4}, \cdots } \right),$

    并令 $T = {T_1} \oplus {T_2}$,则    ${\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.n\left( {T - \lambda I} \right) > \left. {d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:} \right.n( {T - \lambda I} ) = \infty \} \cup \left[ {\rho _a}( T ) \cap\right.$$\left. \sigma ( T ) \right]$${T^ * }$ 不满足 $\left( {{\omega _1}} \right)$ 性质,但 $T$ 不满足 $\left( \omega \right)$ 性质.

    根据定理2,我们可以得到以下推论:

    推论3 若 $T$ 满足 $\left( {{\omega _1}} \right)$ 性质. 则 ${T^ * }$ 满足 $\left( \omega \right)$ 性质且为isoloid算子 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    ${\left[ \rho \right._1}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {{T^ * } - \bar \lambda I} \right)} \right. > \left. {d\left( {{T^ * } - \bar \lambda I} \right)} \right\} \cup \left. {\left\{ {\lambda \in {\bf{C}}:n\left( {{T^ * } - \bar \lambda I} \right)} \right. = \infty } \right\} \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$.

    推论4 设 $T \in B\left( H \right)$. ${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质,$T$ 满足 $\left( \omega \right)$ 性质且 $T$ 为isoloid算子 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    ${\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\cap\left. { {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\rm{iso}}\sigma \left( T \right):n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$.

    证明 因为     $\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\} = \left\{ {\lambda \in {\rm{iso}}\sigma \left( T \right):n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\} \cup $              $\left\{ {\lambda \in {\rm{acc}}\sigma \left( T \right):d\left( {T - \lambda I} \right)} \right. < n( T - \lambda I)=\left. {\infty } \right\} \cup $              $\left\{ {\lambda \in {\rm{acc}}\sigma} \right. \left( T \right): d\left( {T - \lambda I} \right)=n\left( {T - \lambda I} \right)\left. { = \infty } \right\} $,根据定理2可得     ${\sigma _b}\left( T \right) \!=\! {\left[ \rho \right._1}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T \!\!-\! \lambda I} \right)} \right. \!>\!\left. {d\left( {T \!\!-\!\! \lambda I} \right)} \right\} \cup \left. {\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right. \!=\! \infty } \right\} \cup\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq $        $ {\left[ \rho \right._1}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup $        $\left\{ {\lambda \in {\bf{C}}:n\left( {T \!\!-\!\! \lambda I} \right)} \right. > \left. {d\left( {T \!-\! \lambda I} \right)} \right\} \cup\left\{ {\lambda \in {\rm{iso}}\sigma \left( T \right):} \right. \left.n\left( {T - \lambda I} \right)= { \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq $        ${\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup {\left[ \rho \right._1}\left( T \right) \cap {\rho _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right.> $$\left. { d\left( {T - \lambda I} \right) > 0} \right\} \cup $        $\left\{ {\lambda \in {\rm{iso}}\sigma \left( T \right):n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$. 由于 $T$${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质,可得     ${\;\rho _1}\left( T \right) \cap {\rho _w}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right) = \emptyset $ 以及 $\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right.\left. { > d\left( {T - \lambda I} \right) > 0} \right\} = \emptyset $. 从而    ${\sigma _b}\left( T \right) \subseteq {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\cap $$\left. { {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\rm{iso}}\sigma \left. {\left( T \right):n\left( {T - \lambda I} \right) = \infty } \right\}} \right. \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right].$ 其反包含显然成立,因此等式成立.

    反之,由所给条件可得     ${\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\rm{iso}}\sigma \left. {\left( T \right):n\left( {T - \lambda I} \right) = \infty } \right\}} \right. \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq$        $ {\left[ {\partial \rho } \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup {\left[ {\operatorname{int} \rho } \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap {\rm{acc}}\left. {\sigma \left( T \right)} \right] \cup $        $\{ \lambda \in {\rho _1} ( T ):n( {T - \lambda I} ) =d\left( {T - \lambda I} \right) = \infty \} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] \subseteq $        $\partial {\rho _1}\left( T \right) \cup \{ \lambda \in \operatorname{int} {\rho _1} ( T ):d( {T - \lambda I} ) = \infty \} \cup [ {{\rho _a}( T ) \cap \sigma ( T )} ] \cup [ {\rho _s}( T ) \cap \sigma ( T ) ] \subseteq {\sigma _b}\left( T \right). $根据推论2中(1)即可得 $T$${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质. 再者,由     ${{\rm{\pi }}_{00}}\!\left( T \right) \!\cap\! \left\{ {{{\left[ \rho \right.}_1}\!\!\left( T \right) \!\cap\! {\sigma _w}\!\left( T \right)\left. { \!\cap {\rm{acc}}\sigma \!\left( T \right)} \right] \!\cup\! } \right.\left\{ {\lambda \in {\rm{iso}}\sigma } \right.\!\left( T \right)\!:$$\left. {n\left( T \right. \!\!-\!\! \left. {\lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left. {\left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]} \right\} \!=\! \emptyset $,即 ${{\rm{\pi }}_{00}}\left( T \right) \subseteq {\rho _b}\left( T \right) \subseteq {\rho _{ea}}\left( T \right)$. 因此可得 $T$ 满足 $\left( \omega \right)$ 性质.

    ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right)$. 若 $n\left( {T - {\lambda _0}I} \right) = 0$,则 ${\lambda _0} \notin {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in } \right.{\rm{iso}}\sigma \left( T \right):n\left( T \right.\left. {\left. { - \lambda I} \right) = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap } \right.$$\left. {\sigma \left( T \right)} \right] \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right]$. 故有 ${\lambda _0} \in {\rho _b}\left( T \right) \subseteq {\rho _w}\left( T \right)$,即 ${\lambda _0} \in \rho \left( T \right)$,这与 ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right)$ 矛盾. 因此,${\rm{iso}}\sigma \left( T \right) \subseteq $${\sigma _p}\left( T \right)$,即 $T$ 为isoloid算子. 由此,${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质,$T$ 满足 $\left( \omega \right)$ 性质且 $T$ 为isoloid算子得证. 证毕.

    在定理2中,如果我们去掉条件“${T^ * }$ 满足 $\left( {{\omega _1}} \right)$ 性质”,则可得到以下结论.

    定理3 设 $T \in B\left( H \right)$. $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    ${\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \{ \lambda \in {\bf{C}}:n\left( {T - \lambda I} \right) \left. { > d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]$.

    证明 由定理2可得     $ {\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {{\bf{C}}}:n\left( {T - \lambda I} \right)} \right. > \left. {d\left( {T - \lambda I} \right)} \right\} \cup $        $ \left\{ \lambda \right. \in \left. {{\bf{C}}:n\left( {T - \lambda I} \right) = \infty } \right\} \cup\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] =$        $ {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup {\left[ \rho \right._1}\left( T \right) \cap {\rho _w}\left( T \right)\left. { \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right.\left. { > d\left( {T - \lambda I} \right)} \right\} \cup $        $\left\{ \lambda \in {\bf{C}}: \right.n( {T - \lambda I} ) = \infty \} \cup [ {\rho _a}( T ) \cap \sigma ( T ) ]$. 由 $T$ 满足 $\left( \omega \right)$ 性质,可得 ${\rho _1}\left( T \right) \cap {\rho _w}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right) = \emptyset $. 因此    ${\sigma _b}\left( T \right) = $$ {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.n$$\left. {\left( {T - \lambda I} \right) > d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {{\bf{C}}}:n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\} \cup $        $\left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] $.

    反之,由于     $[ {{\sigma _a}( T )\backslash {\sigma _{ea}}( T )} ] \cap \{ { {{[ \rho }_1}( T ) \cap {\sigma _w}( T ) \cap {{\rm{acc}}\sigma ( T )} ]} \cup \{ {\lambda \in {{\bf{C}}}:n( {T - \lambda I} )} > d( T - { {\lambda I} )} \} \cup $    $\{ {\lambda \in {\bf{C}}:n( {T - \lambda I} )= }{ \infty } \} \cup [ {{\rho _a}( T ) \cap \sigma ( T )} ] \} = \emptyset $,故 ${\sigma _a}\left( T \right)\backslash {\sigma _{ea}}\left( T \right) \subseteq {\rho _b}\left( T \right)$,即 ${\sigma _a}\left( T \right)\backslash {\sigma _{ea}}\left( T \right) \subseteq {{\rm{\pi }}_{00}}\left( T \right)$. 此外,设 ${\lambda _0} \in {{\rm{\pi }}_{00}}\left( T \right)$. 若 ${\lambda _0} \in \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)}> \right. d\left( {T - } \right. $$\left. {\left. {\lambda I} \right)} \right\}$,则 ${\lambda _0} \in {\rho _b}\left( T \right)$. 若 ${\lambda _0} \notin \left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} \right.\left. { > d\left( {T - \lambda I} \right)} \right\}$,则     $ {\lambda _0} \notin {[ \rho _1}( T ) \cap {\sigma _w}( T ) \cap $$ {{\rm{acc}}\sigma ( T )} ] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T \!-\! \lambda I} \right)} \right.$$\left. { > d\left( {T \!-\! \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T\! -\! \lambda I} \right)} \right.\left. { = \infty } \right\} \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]$,即 ${\lambda _0} \in {\rho _b}\left( T \right) \subseteq {\rho _{ea}}\left( T \right)$. 因此 ${{\rm{\pi }}_{00}}\left( T \right)$$ \subseteq {\sigma _a}\left( T \right)\backslash {\sigma _{ea}}\left( T \right)$. 并且,我们知道     $\left\{ {\lambda \!\in\! {\rm{iso}}\sigma \left( T \right):n\left( {T \!-\! \lambda I} \right) \!=\! 0} \right\} \!\cap\! \left\{ {{{\left[ \rho \right.}_1}\left( T \right) \!\cap\! {\sigma _w}\left( T \right) \!\cap\! \left. {{\rm{acc}}\sigma \left( T \right)} \right]} \right. \!\cup\! \left\{ {\lambda \!\in\! {{\bf{C}}}:n\left( {T \!-\! \lambda I} \right)} \right. \! >\left. { d\left( {T - \lambda I} \right)} \right\} \cup $    $ \left\{ {\lambda \in {{\bf{C}}}:n\left( {T - \lambda I} \right)} \right.\left. { = \infty } \right\}\left. { \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right]} \right\}{\rm{ = }}\emptyset $,由此即可得 $\left\{ {\lambda \in {\rm{iso}}\sigma \left( T \right):n\left( {T - \lambda I} \right) = 0} \right\} \subseteq {\rho _b}\left( T \right)$. 在这种情况下,$\left\{ {\lambda \in {\rm{iso}}\sigma \left( T \right):n\left( {T - \lambda I} \right) = 0} \right\} = \emptyset $. 因此 ${\rm{iso}}\sigma \left( T \right) \subseteq {\sigma _p}\left( T \right)$. 这就证明了 $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子. 证毕.

    对于 $T$ 的共轭算子 ${T^ * }$,类似定理3的证明,可得以下推论.

    推论5 设 $T \in B\left( H \right)$. ${T^ * }$ 满足 $\left( \omega \right)$ 性质且为isoloid算子 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    $ {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:n\left( {T^ * }\!-\! \right.} \right.\left.\bar \lambda I \right)\left. {> d\left( {{T^ * } \!-\! \bar \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:n\left( {{T^ * } \!-\! \bar \lambda I} \right)} \right.\left. { \!=\! \infty } \right\} \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] $.

    推论6 设 $T \in B\left( H \right)$. $T$${T^ * }$ 均满足 $\left( \omega \right)$ 性质且均为isoloid算子 $ \Leftrightarrow {\sigma _b}\left( T \right) = $    ${\left[ \rho \right._1}\left( T \right) \cap \left. {{\sigma _w}\left( T \right) \cap {\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ \lambda \in {\bf{C}}:n\left( {T - \lambda I} \right) = n\left. {\left( {{T^ * } - \bar \lambda I} \right) = \infty } \right\} \right. \cup \left[ {{\rho _a}\left( T \right) \cap \sigma \left( T \right)} \right] \cup \left[ {{\rho _s}\left( T \right) \cap \sigma \left( T \right)} \right] $.

    推论7 设 $T \in B\left( H \right)$. $T$ 满足 $\left( \omega \right)$ 性质 $ \Leftrightarrow {\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {{\bf{C}}}:} \right.\left. {n\left( {T - \lambda I} \right) > d\left( {T - \lambda I} \right)} \right\} \cup $$\left\{ {\lambda \in {\bf{C}}:n\left( {T - \lambda I} \right)} { = \infty } \right\} \cup \left\{ {\lambda \in \sigma \left( T \right):n\left( {T - \lambda I} \right) = 0} \right\} $.

    在文献[15]中,介绍了如下引理.

    引理1 设 $T \in B\left( H \right)$. 若 $K \in B\left( H \right)$ 是一个可交换的幂有限秩算子,则:

    (1)$K$ 是一个Riesz算子;

    (2)$n\left( {T + K} \right) < \infty \Leftrightarrow n\left( T \right) < \infty $

    (3)${\rm{iso}}\sigma \left( {T + K} \right) \subseteq {\rm{iso}}\sigma \left( T \right) \cup \rho \left( T \right)$

    (4)${\rm{iso}}{\sigma _a}\left( {T + K} \right) \subseteq {\rm{iso}}{\sigma _a}\left( T \right) \cup {\rho _a}\left( T \right)$.

    显然,若 $T$ 是一个有限秩算子,则 $T$ 是一个幂有限秩算子.

    定理4 设 $T \in B\left( H \right)$. 则以下命题等价:

    (1) ${\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.\left. {n\left( {T - \lambda I} \right) > d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:} \right.n( T - \left. {\lambda I) = \infty } \right\}$

    (2) 对任意可交换的有限秩算子 $F \in B\left( H \right)$,有 ${\sigma _a}\left( T \right) = \sigma \left( T \right)$,并有 $T + F$ 满足 $\left( \omega \right)$ 性质且为isoloid算子.

    证明 $ (1) \Rightarrow (2)$. 由定理3可知 $T$ 满足 $\left( \omega \right)$ 性质且为isoloid算子. 设 ${\lambda _0} \in {\rho _a}\left( T \right) \cap \sigma \left( T \right)$,则有 ${\lambda _0} \notin {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.\left. {n\left( {T - \lambda I} \right) > d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:} \right.n\left( T \right. - \left. {\lambda I} \right) = \left. \infty \right\}$,故 ${\lambda _0} \in {\rho _b}\left( T \right)$. 这与 ${\lambda _0} \in {\rho _a}\left( T \right) \cap \sigma \left( T \right)$ 矛盾,所以 $\sigma \left( T \right) \subseteq {\sigma _a}\left( T \right)$. 又显然有 $\sigma \left( T \right) \supseteq {\sigma _a}\left( T \right)$,因此 ${\sigma _a}\left( T \right) = \sigma \left( T \right)$. 此外,设 ${\lambda _0} \in {\sigma _a}\left( {T + F} \right)\backslash {\sigma _{ea}}\left( {T + F} \right)$. 若 ${\lambda _0} \in {\rho _a}\left( T \right)$,则由 ${\sigma _a}\left( T \right) = \sigma \left( T \right)$ 可得 ${\lambda _0} \in \rho \left( T \right) \subseteq {\rho _b}\left( T \right) = {\rho _b}\left( {T + F} \right)$,即 ${\lambda _0} \in $${{\rm{\pi }}_{00}}\left( {T + F} \right) $;若 ${\lambda _0} \in {\sigma _a}\left( T \right)$,则 ${\lambda _0} \in {\sigma _a}\left( T \right)\backslash {\sigma _{ea}}\left( T \right)$. 由 $T$ 满足 $\left( \omega \right)$ 性质,有 ${\lambda _0} \in {\rho _b}\left( T \right) = {\rho _b}\left( {T + F} \right)$,即 ${\lambda _0} \in {{\rm{\pi }}_{00}}\left( {T + F} \right)$. 故 ${\sigma _a}\left( {T + F} \right)\backslash {\sigma _{ea}}\left( {T + F} \right) \subseteq {{\rm{\pi }}_{00}}\left( {T + F} \right)$. 设 ${\lambda _0} \in {{\rm{\pi }}_{00}}\left( {T + F} \right)$,则 ${\lambda _0} \in {\rm{iso}}\sigma \left( {T + F} \right)$$n\left( {T + F - {\lambda _0}I} \right) < \infty $. 由引理1可得 ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right) \cup \rho \left( T \right)$$n\left( {T - {\lambda _0}I} \right) < \infty $. 若 ${\lambda _0} \in \rho \left( T \right)$,则 ${\lambda _0} \in {\rho _b}\left( T \right) \subseteq {\rho _{ea}}\left( T \right) = {\rho _{ea}}\left( {T + F} \right)$;若 ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right)$,则由 $T$ 是isoloid算子可得 ${\lambda _0} \in {{\rm{\pi }}_{00}}\left( T \right)$,即 ${\lambda _0} \in {\rho _{ea}}\left( T \right) = {\rho _{ea}}\left( {T + F} \right)$. 因此 ${{\rm{\pi }}_{00}}\left( {T + F} \right) \subseteq {\sigma _a}\left( T \right.\left. { + F} \right)\backslash {\sigma _{ea}}\left( {T + F} \right)$. 综上,这就证明了 $T + F$ 满足 $\left( \omega \right)$ 性质.

    接下来,我们证明 $T + F$ 是isoloid算子. 设 ${\lambda _0} \in {\rm{iso}}\sigma \left( {T + F} \right)$,则由引理1可得 ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right)$$ \cup \rho \left( T \right)$. 显然,$\;\rho \left( T \right) \subseteq {\rho _b}\left( {T + F} \right)$. 因此,若 ${\lambda _0} \in \rho \left( T \right)$,则有 $n\left( {T + F - {\lambda _0}I} \right) > 0$. 若不然,则有 ${\lambda _0} \in \rho \left( {T + F} \right)$,这与 ${\lambda _0} \in $$ {\rm{iso}}\sigma \left( {T + F} \right)$ 矛盾. 若 ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right)$,可以证明 $n\left( {T + F - {\lambda _0}I} \right) > 0$. 实际上,若 $n\left( {T + F - {\lambda _0}I} \right) = 0$,则由 $T$ 是isoloid算子可得 ${\lambda _0} \in {{\rm{\pi }}_{00}}\left( T \right)$. 此外,由 $T$ 满足 $\left( \omega \right)$ 性质,则有 ${\lambda _0} \in {\rho _b}\left( {T + F} \right)$. 故 $T + F - {\lambda _0}I$ 可逆,矛盾. 因此 $T + F$ 是isoloid算子.

    $(2) \Rightarrow (1) $. 设 ${\lambda _0} \notin {\left[ \rho \right._1}\left( T \right) \cap {\sigma _w}\left( T \right) \cap \left. {{\rm{acc}}\sigma \left( T \right)} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.\left. {n\left( {T - \lambda I} \right) > d\left( {T - \lambda I} \right)} \right\} \cup \left\{ {\lambda \in {\bf{C}}:} \right.$$n\left( T \right. - \left. {\lambda I} \right) = \left. \infty \right\}$. 则 $n\left( {T - {\lambda _0}I} \right) < \infty $$n\left( {T - {\lambda _0}I} \right) \leqslant d\left( {T - {\lambda _0}I} \right)$.

    ${\lambda _0} \notin {\rho _1}\left( T \right)$,则有 ${\lambda _0} \in {\rho _{ea}}\left( T \right)$,即 ${\lambda _0} \in {\rho _a}\left( T \right) \cup {{\text{π}}_{00}}\left( T \right)$. 由 ${\sigma _a}\left( T \right) = \sigma \left( T \right)$ 可知 ${\lambda _0} \in {\rho _b}\left( T \right)$. 这与 ${\lambda _0} \in {\sigma _1}\left( T \right)$ 矛盾;若 ${\lambda _0} \notin {\sigma _w}\left( T \right)$,类似以上证明可得 ${\lambda _0} \notin {\sigma _b}\left( T \right)$;若 ${\lambda _0} \notin {\rm{acc}}\sigma \left( T \right)$,则 ${\lambda _0} \in {\rm{iso}}\sigma \left( T \right) \cup \rho \left( T \right)$. 由于 $T$ 是isoloid算子,则可得 ${\lambda _0} \notin {\sigma _b}\left( T \right)$. 证毕.

    推论8 设 $T \in B\left( H \right)$. 则以下条件等价:

    (1)${\sigma _b}\left( T \right) = {\left[ \rho \right._1}\left( T \right) \!\cap\! {\sigma _w}\left( T \right) \cap \left. {\left\{ {\lambda \in {\rm{acc}}{\sigma _b}\left( T \right):d\left( {T - \lambda I} \right) = \infty } \right\}} \right] \cup \left\{ {\lambda \in {\bf{C}}:} \right.n\left( {T - \lambda I} \right) > d\left( T \right. - {\left. {\lambda I} \right\} \cup }$      $ \left\{ {\lambda \in {\bf{C}}:}n ( T - \right.{\lambda I} )\! =\! \infty\} $

    (2)对任意可交换的有限秩算子 $F \in B\left( H \right)$,有 ${\sigma _a}\left( T \right) = \sigma \left( T \right)$,且 $\left\{ {\lambda \in {\rm{iso}}} \right.{\sigma _b}\left( T \right):n\left( T \right. - \left. {\lambda I} \right)$$\left. { < \infty } \right\} = \emptyset $,并有 $T + F$ 满足 $\left( \omega \right)$ 性质且为isoloid算子.

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