本文中,设$X$是无限维复Banach空间, $L(X)$表示从$X$到$X$的有界线性算子的全体.对于$T\in L(X)$, $T^*$表示$T$的共轭算子, $\alpha(T)$表示零空间$N(T)$的维数, $\beta(T)$表示值域$R(T)$的亏维. $p(T)$和$q(T)$分别表示$T$的升指数和降指数,即

$p(T)=\inf\{n\in{\Bbb N}: N(T^n)=N(T^{n+1})\}, $

$q(T)=\inf\{n\in{\Bbb N}: R(T^n)=R(T^{n+1})\}$

(若下确界不存在,记$p(T)=\infty$, $q(T)=\infty$).若$T$的升指数和降指数都是有限的,则它们相等(见文献[1]的定理1.19).若$\alpha(T)$和$\beta(T)$都有限($R(T)$闭且$\alpha(T)<\infty$),则称$T$是Fredholm算子(上半Fredholm算子). $T$的Fredholm指标定义为$ind(T)=\alpha(T)-\beta(T)$.若$T$是Fredholm算子且$p(T)=q(T)<\infty$ (上半Fredholm算子且$p(T)<\infty$),则称$T$是Browder算子(上半Browder算子).指标为0的Fredholm算子称为Weyl算子.若$p(T)<\infty$且$R(T^{p(T)+1})$闭,则称$T$为左Drazin可逆算子.若$q(T)<\infty$且$R(T^{q(T)})$闭,则称$T$为右Drazin可逆算子.若$p(T)=q(T)<\infty$,则称$T$为Drazin可逆算子.显然既左Drazin可逆又右Drazin可逆的算子是Drazin可逆算子.

对任意的$n\in{\Bbb N}$, $T_n$表示$T$在$R(T^n)$上的限制,即$T: R(T^n)\rightarrow R(T^n)$(特别地, $T_0=T$).若存在$n\in{\Bbb N}$,使得$R(T^n)$是闭的且$T_n$是Fredholm算子(上半Fredholm算子),则称$T$是B-Fredholm算子(上半B-Fredholm算子).由文献[2]的定理2.1知,若存在$n\in{\Bbb N}$使得$R(T^n)$是闭的且$T_n$是上半Fredholm算子,则$R(T^m)$是闭的, $T_m$为上半Fredholm算子且$ind(T_m)=ind(T_n)$, $\forall m\geq n$.所以,半B-Fredholm算子$T$的指标可以用半Fredholm算子$T_n$的指标定义,即$ind(T)=ind(T_n)$.若$T$是B-Fredholm算子且$ind(T)=ind(T_n)=0$,则称$T$为B-Weyl算子.

对于$T\in L(X)$,记$T$的谱为$\sigma(T)$,近似点谱为$\sigma_a(T)$,点谱为$\sigma_p(T)$,以及下面各种谱

$ \begin{eqnarray*}\mbox{ Weyl谱:}&&\sigma_{w}(T)=\{\lambda\in{\Bbb C}: \lambda I-T\ \mbox{不是Weyl算子\};} \\\mbox{上半Browder谱:}&&\sigma_{ub}(T)=\{\lambda\in{\Bbb C}: \lambda I-T\ \mbox{不是上半Browder算子\};} \\\mbox{ Browder谱:} &&\sigma_{b}(T)=\{\lambda\in{\Bbb C}: \lambda I-T\ \mbox{不是Browder算子\};} \\\mbox{ B-Weyl谱:} &&\sigma_{bw}(T)=\{\lambda\in{\Bbb C}: \lambda I-T\ \mbox{不是B-Weyl算子\};} \\ \mbox{ Drazin谱:}&& \sigma_{d}(T)=\{\lambda\in{\Bbb C}: \lambda I-T\ \mbox{不是Drazin可逆算子\}.} \end{eqnarray*}$

若$\lambda I-T$是左Drazin可逆,则称$\lambda\in\sigma_a(T)$为$T$的左极点,记$\Pi_a(T)$为$T$的所有左极点构成的集合.若

$0<p(\lambda I-T)=q(\lambda I-T)<\infty$

则称$\lambda\in{\Bbb C}$为$T$的预解式的极点(简称极点),记$\Pi(T)$为$T$的所有极点所组成的集合.同时,分别用$\Pi^0(T)$和$\Pi^0_a(T)$表示$T$的所有有限秩的极点和有限秩的左极点,即

$\Pi^0(T)=\{\lambda\in\Pi(T): \alpha(\lambda I-T)<\infty\}, \Pi^0_a(T)=\{\lambda\in\Pi_a(T):\alpha(\lambda I-T)<\infty\}.$

此外,记

$E(T)=\{\lambda\in iso\sigma(T): 0<\alpha(\lambda I-T)\}; E_a(T)=\{\lambda\in iso\sigma_a(T): 0<\alpha(\lambda I-T)\};$

$E^0(T)=\{\lambda\in E(T): \alpha(\lambda I-T)<\infty\}; E^0_a(T)=\{\lambda\in E_a(T): \alpha(\lambda I-T)<\infty\}.$

显然,有结论: $\Pi(T)\subseteq E(T); \Pi_a(T)\subseteq E_a(T); \Pi^0(T)\subseteq E^0(T); \Pi^0_a(T)\subseteq E^0_a(T); E(T)\subseteq E_a(T); $$E^0(T)\subseteq E^0_a(T); \Pi(T)\subseteq\Pi_a(T)$和$\Pi^0(T)\subseteq\Pi^0_a(T)$.以及$\Pi^0(T)=\sigma(T)\backslash\sigma_b(T), $$ \Pi^0_a(T)=\sigma_a(T)\backslash\sigma_{ub}(T)$.

如果$\sigma(T)\setminus\sigma_{w}(T)= E^{0}(T)$,则称$T$满足Weyl定理^[3].如果$\sigma(T)\setminus\sigma_{w}(T)=\Pi^{0}(T)$,则称$T$满足Browder定理^[4].利用B-Fredholm理论, M.Berkani和J.J.Koliha在文献[5]中对Weyl定理进行了推广,如果$\sigma(T)\setminus\sigma_{bw}(T)=E(T)$,则称$T$满足广义Weyl定理;如果$\sigma(T)\setminus\sigma_{bw}(T)=\Pi(T)$,则称$T$满足广义Browder定理.且由文献[5-7]可知

$\mbox{广义Weyl定理$\Longrightarrow$Weyl定理$\Longrightarrow$Browder定理$\Longleftrightarrow$广义Browder定理.}$

近年来,许多学者对Weyl型定理进行了变形和推广,并对其进行了深入研究.本文新定义了两种谱性质:性质$(H)$和性质$(gH)$.若$\sigma(T)\setminus\sigma_{bw}(T)=E^0_a(T)$,则称$T$满足性质$(H)$;若$\sigma(T)\setminus\sigma_{bw}(T)=\Pi^0_a(T)$,则称$T$满足性质$(gH)$.并探讨了这两种谱性质之间的关系以及同其它Weyl型定理之间的关系.最后研究了这两种谱性质在可交换的幂零算子、拟幂零算子、有限秩算子和Riesz算子摄动下的稳定性.

我们先引入以下符号,对任意$n\in{\Bbb N}$,令

$c_n(T)=\dim(R(T^n)/R(T^{n+1}));$

$c'_n(T)=\dim(N(T^{n+1})/N(T^n));$

$k_n(T)=\dim[(R(T^n)\cap N(T))/(R(T^{n+1})\cap N(T))].$

设$T\in L(X)$, $d\in{\Bbb N}$,如果

$R(T)+N(T^n)=R(T)+N(T^d), \forall n\geq d, $

则称$T$对$n\geq d$有一致降指数.如果还满足$R(T)+N(T^d)$是闭的,则称$T$对$n\geq d$有拓扑一致降指数.

Grabiner在文献[8]中引入了有拓扑一致降指数的算子,它是Kato型算子的一个推广,自然也涵盖了Fredholm理论中的许多算子,如半Fredholm算子, Kato型算子,拟Fredholm算子,半B-Fredholm算子等.而文献[8]的定理4.7是关于有拓扑一致降指数算子的重要定理.

Dunford在文献[9-10]中引入了单值扩张性的概念(简称SVEP),它作为研究算子的谱的有力工具极大地丰富了算子谱结构的经典研究.如果对$\lambda_0$的任何开邻域$U$,满足式子$(\lambda I-T)f(\lambda)=0 (\forall\lambda\in U)$的唯一解析函数$f:U\longrightarrow X$是$U$上的零函数,则称$T$在$\lambda_0\in{\Bbb C}$处有SVEP.若$T$在任意的$\lambda\in{\Bbb C}$处都有SVEP,则称$T$有SVEP.显然, $T$和$T^*$在$\lambda\in iso\sigma(T)$处都有SVEP,且$T$在$\lambda\in iso\sigma_a(T)$处有SVEP.

定义2.1  设$T\in L(X)$,若$T$满足$\sigma(T)\setminus\sigma_{bw}(T)=E^0_a(T)$,则称$T$满足性质$(H)$.若$T$满足$\sigma(T)\setminus\sigma_{bw}(T)=\Pi^0_a(T)$,则称$T$满足性质$(gH)$.

引理2.1^[11]  设$T$是上半B-Fredholm算子,若$\alpha(T)<\infty$,则$T$是上半Fredholm算子.

定理2.1  设$T\in L(X)$,若$T$满足性质$(H)$,则$T$满足性质$(gH)$.

证  设$T$满足性质$(H)$,则$\sigma(T)\setminus\sigma_{bw}(T)=E^0_a(T)$.显然$\Pi^0_a(T)\subseteq E^0_a(T)=\sigma(T)\setminus\sigma_{bw}(T)$.下证$\sigma(T)\setminus\sigma_{bw}(T)\subseteq\Pi^0_a(T)$.对任意的$\lambda\in\sigma(T)\setminus\sigma_{bw}(T)$,有$\lambda I-T$是B-Weyl算子.由于$T$满足性质$(H)$,则$\lambda\in E^0_a(T)$,即$\lambda\in iso\sigma_a(T)$且$0<\alpha(\lambda I-T)<\infty$.因$\lambda I-T$是B-Fredholm算子且$0<\alpha(\lambda I-T)<\infty$,由引理2.1知$\lambda I-T$是上半Fredholm算子.又因$\lambda\in iso\sigma_a(T)$,则$T$在$\lambda$处有SVEP.于是,由文献[1]的定理2.45知$p(\lambda I-T)<\infty$,则$\lambda I-T$是上半Browder算子,从而$\lambda\in\sigma_a(T)\setminus\sigma_{ub}(T)=\Pi^0_a(T)$,即$\sigma(T)\setminus\sigma_{bw}(T)\subseteq\Pi^0_a(T)$.因此, $T$满足性质$(gH)$.

下面例子表明,通常情况下,定理2.1的逆命题不成立.

例2.1  设$T\in L(l^2({\Bbb N}))$定义为

$T(x_1, x_2, x_3, \cdots)=(\frac{1}{3}x_2, \frac{1}{4}x_3, \cdots), (x_n)\in l^2({\Bbb N}), $

则$\sigma(T)=\sigma_{bw}(T)=\{0\}, E^0_a(T)=\{0\}$且$\Pi^0_a(T)=\emptyset$,所以$\sigma(T)\setminus\sigma_{bw}(T)=\emptyset=\Pi^0_a(T), $$\sigma(T)\setminus\sigma_{bw}(T)\neq E^0_a(T)$,即$T$满足性质$(gH)$,但不满足性质$(H)$.

性质$(H)$和性质$(gH)$有下面重要结论.

定理2.2  设$T\in L(X)$,那么

$(1)$若$T$满足性质$(H)$,则$E^0(T)=E^0_a(T)$;

$(2)$若$T$满足性质$(gH)$,则$\Pi^0(T)=\Pi^0_a(T)$.

证  (1) 设$T$满足性质$(H)$,则$\sigma(T)\setminus\sigma_{bw}(T)=E^0_a(T)$.对任意的$\lambda\in E^0_a(T)$,有$\lambda\in\sigma(T)\setminus\sigma_{bw}(T)$,即$\lambda I-T$是B-Weyl算子,则存在$d\in{\Bbb N}$,使得$\lambda I-T$对任意的$n\geq d$有拓扑一致降指数,由文献[8]的定理4.7知,存在$\lambda$的去心邻域$U_1$,使得对$\forall n\geq d$和$\mu\in U_1$,有

$\alpha(\mu I-T)=c'_0(\mu I-T)=c'_n(\lambda I-T)\ \mbox{和}\ \beta(\mu I-T)=c_0(\mu I-T)=c_n(\lambda I-T).$

因$\lambda I-T$是B-Weyl算子,则当$n$充分大时,有$c_n(\lambda I-T)=c'_n(\lambda I-T)$,从而

$\alpha(\mu I-T)=c'_n(\lambda I-T)=c_n(\lambda I-T)=\beta(\mu I-T).$

又由于$\lambda\in E^0_a(T)$,则存在$\lambda$的去心邻域$U_2$,使得对$\forall\mu\in U_2(\mu\neq\lambda)$有$\alpha(\mu I-T)=0$.令$U=U_1\cap U_2$,则对$\forall\mu\in U$有$\alpha(\mu I-T)=\beta(\mu I-T)=0$,即$\mu I-T$可逆,所以$\lambda\in iso\sigma(T)$,从而$\lambda\in E^0(T)$.所以$E^0_a(T)\subseteq E^0(T)$.又$E^0(T)\subseteq E^0_a(T)$对任意的$T\in L(X)$都成立.因此, $E^0(T)=E^0_a(T)$.

(2)  设$T$满足性质$(gH)$,则$\sigma(T)\setminus\sigma_{bw}(T)=\Pi^0_a(T)$.对任意的$\lambda\in\Pi^0_a(T)$,有$\lambda\in\sigma(T)\setminus\sigma_{bw}(T)$,则$\lambda I-T$是B-Weyl算子,即存在$n\in{\Bbb N}$,使得$R((\lambda I-T)^n)$是闭的, $(\lambda I-T)_n$是Fredholm算子且$ind((\lambda I-T)_n)=0$.因$\alpha(\lambda I-T)<\infty$,由引理2.1知$\lambda I-T$是上半Fredholm算子,则$R(\lambda I-T)$是闭的且$ind(\lambda I-T)=ind((\lambda I-T)_n)=0$,即$\lambda I-T$是Weyl算子.又因$\lambda\in\Pi^0_a(T)\subseteq\Pi_a(T)$,有$p(\lambda I-T)<\infty$.由文献[1]的定理1.21知$q(\lambda I-T)=p(\lambda I-T)<\infty$,即$\lambda\in\Pi(T)$.因$\lambda\in\sigma(T)$且$\alpha(\lambda I-T)<\infty$,所以$\lambda\in\Pi^0(T)$,即$\Pi^0_a(T)\subseteq\Pi^0(T)$.又$\Pi^0(T)\subseteq\Pi^0_a(T)$对任意的$T\in L(X)$都成立.因此, $\Pi^0(T)=\Pi^0_a(T)$.

例2.1表明通常情况下性质$(H)$和性质$(gH)$是不等价的,然而,当$T$满足下面定理2.3的条件时,利用定理2.1和2.2容易证明它们是等价的.

定理2.3  设$T\in L(X)$,则下列叙述等价

$(1)$$T$满足性质$(H)$;

$(2)$$T$满足性质$(gH)$且$E^0_a(T)=\Pi^0_a(T)$;

$(3)$$T$满足性质$(gH)$且$E^0_a(T)=\Pi^0(T)$.

设$T\in L(X)$,称$T$是a-polaroid算子,若$iso\sigma_a(T)\subseteq\Pi(T)$.若$T$是a-polaroid算子,可进一步得到性质$(H)$和性质($gH)$等价的结论.

定理2.4  设$T$是a-polaroid算子, $T$满足性质$(H)$当且仅当$T$满足性质$(gH)$.

证  由定理2.3知,只需证明: $E^0_a(T)=\Pi^0(T)$.对任意的$T\in L(X)$,显然$\Pi^0(T)\subseteq E^0(T)\subseteq E^0_a(T)$.反之,对任意的$\lambda\in E^0_a(T)$,有$\lambda\in iso\sigma_a(T)$且$0<\alpha(\lambda I-T)<\infty$.由于$T$是a-polaroid算子,则$\lambda\in\Pi(T)$.因$\alpha(\lambda I-T)<\infty$,则$\lambda\in\Pi^0(T)$.因此, $E^0_a(T)=\Pi^0(T)$.

下面探讨性质$(H)$和$(gH)$与Weyl定理、Browder定理、广义Weyl定理和广义Browder定理之间的关系.

定理2.5  设$T\in L(X)$,若$T$满足性质$(H)$,则$T$也满足Weyl定理且$\sigma(T)=\sigma_{bw}(T)\cup iso\sigma(T)$.

证  若$T$满足性质$(H)$,则$\sigma(T)\setminus\sigma_{bw}(T)=E^0_a(T)$.对任意的$\lambda\in\sigma(T)\setminus\sigma_w(T)$,有$\lambda\in\sigma(T)\setminus\sigma_{bw}(T)$,则$\lambda\in E^0_a(T)$.因$T$满足性质$(H)$,由定理2.2可知$E^0(T)=E^0_a(T)$,所以$\lambda\in E^0(T)$,即$\sigma(T)\setminus\sigma_w(T)\subseteq E^0(T)$.反之,对任意的$\lambda\in E^0(T)\subseteq E^0_a(T)$,由于$T$满足性质$(H)$,有$\lambda\in\sigma(T)\setminus\sigma_{bw}(T)$,则$\lambda I-T$是B-Weyl算子,即存在$n\in{\Bbb N}$,使得$R((\lambda I-T)^n)$是闭的, $(\lambda I-T)_n$是Fredholm算子且$ind((\lambda I-T)_n)=0$.因$\alpha(\lambda I-T)<\infty$,由引理2.1知$\lambda I-T$是上半Fredholm算子,则$R(\lambda I-T)$是闭的且$ind(\lambda I-T)=ind((\lambda I-T)_n)=0$,即$\lambda I-T$是Weyl算子.所以$\lambda\in\sigma(T)\setminus\sigma_w(T)$,即$E^0(T)\subseteq\sigma(T)\setminus\sigma_w(T)$.因此$T$满足Weyl定理.

因$T$满足性质$(H)$且由定理2.2知, $\sigma(T)\setminus\sigma_{bw}(T)=E^0_a(T)=E^0(T)\subseteq iso\sigma(T), $即$\sigma(T)\subseteq\sigma_{bw}(T)\cup iso\sigma(T)$.又$\sigma_{bw}(T)\cup iso\sigma(T)\subseteq\sigma(T)$显然成立.因此, $\sigma(T)=\sigma_{bw}(T)\cup iso\sigma(T)$.

下面例子表明,通常情况下,定理2.5的逆命题不成立.

例2.2  设$T\in L(l^2({\Bbb N}))$定义为

$T(x_1, x_2, x_3, \cdots)=(0, x_2, x_3, \cdots), (x_n)\in l^2({\Bbb N}), $

则$\sigma(T)=\sigma_a(T)=\{0, 1\}, \sigma_w(T)=\{1\}, $$\sigma_{bw}(T)=\emptyset, $$ E^0(T)=E^0_a(T)=\{0\}$.所以, $\sigma(T)\setminus\sigma_w(T)=\{0\}=E^0(T), $$\sigma(T)\setminus\sigma_{bw}(T)=\{0, 1\}\neq E^0_a(T)$,即$T$满足Weyl定理,但是不满足性质$(H)$.

定理2.6  设$T\in L(X)$,若$T$满足性质$(gH)$,则$T$也满足广义Browder定理且$\sigma(T)=\sigma_{bw}(T)\cup iso\sigma(T)$.

证  若$T$满足性质$(gH)$,则$\sigma(T)\setminus\sigma_{bw}(T)=\Pi^0_a(T)$.对任意的$\lambda\notin\sigma_{bw}(T)$,若$\lambda\in\sigma(T)$,则$\lambda\in\sigma(T)\setminus\sigma_{bw}(T)=\Pi^0_a(T)$,从而$\lambda\in iso\sigma_a(T)$,所以$T$在$\lambda$处有SVEP.若$\lambda\notin\sigma(T)$,显然$T$在$\lambda$处有SVEP.因此,由文献[12]的定理3.2知$T$满足广义Browder定理.

因$T$满足性质$(gH)$且由定理2.2知$\sigma(T)\setminus\sigma_{bw}(T)=\Pi^0_a(T)=\Pi^0(T)\subseteq iso\sigma(T)$,即$\sigma(T)\subseteq\sigma_{bw}(T)\cup iso\sigma(T)$.又$\sigma_{bw}(T)\cup iso\sigma(T)\subseteq\sigma(T)$显然成立.因此, $\sigma(T)=\sigma_{bw}(T)\cup iso\sigma(T)$.

推论2.1  设$T\in L(X)$,若$T$满足性质$(gH)$,则$T$也满足Browder定理.

下面例子表明,通常情况下,定理2.6和推论2.1的逆命题不成立.

如在例2.2中定义的$T\in L(l^2({\Bbb N}))$,有$\sigma(T)=\sigma_a(T)=\{0, 1\}, \sigma_w(T)=\{1\}, $$\sigma_{bw}(T)=\emptyset, $$ \Pi^0(T)=\Pi^0_a(T)=\{0\}$,所以$\sigma(T)\setminus\sigma_{w}(T)=\{0\}=\Pi^0(T), \sigma(T)\setminus\sigma_{bw}(T)=\{0, 1\}\neq\Pi^0_a(T)$,即$T$满足Browder定理,进而也满足广义Browder定理,但是不满足性质$(gH)$.

利用定理2.1和2.6,容易证明加上某些条件后,性质$(H)$和性质$(gH)$可以等价于Browder定理和广义Browder定理,即有下面定理2.7和定理2.8成立.

定理2.7  设$T\in L(X)$,则下列叙述等价

$(1)$$T$满足性质$(H)$; $(2)$$T$满足Browder定理且$\Pi(T)=E^0_a(T)$; $(3)$$T$满足广义Browder定理且$\Pi(T)=E^0_a(T)$.

定理2.8  设$T\in L(X)$,则下列叙述等价

$(1)$$T$满足性质$(gH)$;

$(2)$$T$满足Browder定理且$\Pi(T)=\Pi^0_a(T)$;

$(3)$$T$满足广义Browder定理且$\Pi(T)=\Pi^0_a(T)$.

下面例子表明,通常情况下,性质$(H)$与广义Weyl定理无关.

例2.3  设$Q\in L(l^1)$定义为

$ Q(x_1, x_2, \cdots, x_k, \cdots)=(0, \alpha_1x_1, \alpha_2x_2, \cdots, \alpha_{k-1}x_{k-1}, \cdots), \ (x_n)\in l^1, $

其中$\alpha_i\in{\Bbb C}, 0<|\alpha_i|\leq 1$,且$\sum|\alpha_i|<\infty$.由文献[5]的例$3.12$知, $\overline{R(Q^n)}\neq R(Q^n), $$n=1, 2, \cdots.$考虑$T\in L(l^1\oplus l^1): T=Q\oplus 0$,则$N(T)=\{0\}\oplus l^1, $$\sigma(T)=\sigma_a(T)=\{0\}, $$ E(T)=\{0\}, $$ E^0_a(T)=\emptyset$.由于$R(T^n)=R(Q^n)\oplus\{0\}$,则对$\forall n\in{\Bbb N}, R(T^n)$不闭,从而$\sigma_{bw}(T)=\{0\}$.所以$\sigma(T)\setminus\sigma_{bw}(T)=\emptyset=E^0_a(T), \sigma(T)\setminus\sigma_{bw}(T)\neq E(T)$,即$T$满足性质$(H)$,但是不满足广义Weyl定理.

如在例2.2中定义的$T\in L(l^2(N))$,有$\sigma(T)=\sigma_a(T)=\{0, 1\}, \sigma_{bw}(T)=\emptyset$,以及$E^0(T)=E^0_a(T)=\{0\}$,由于$0<\alpha(I-T)=\infty$,且$E^0(T)\subseteq E(T)$,则$E(T)=\{0, 1\}$.所以$\sigma(T)\setminus\sigma_{bw}(T)=E(T), \sigma(T)\setminus\sigma_{bw}(T)\neq E^0_a(T)$,即$T$满足广义Weyl定理,但是不满足性质$(H)$.

设$T\in L(X)$,若$iso\sigma(T)\subseteq\Pi(T)$,则称$T$是polaroid算子.若$T$是polaroid算子,则性质$(H)$和$(gH)$与广义Weyl定理之间有下面关系.

定理2.9  设$T\in L(X)$是polaroid算子,那么

$(1)$若$T$满足性质$(H)$,则$T$满足广义Weyl定理;

$(2)$若$T$满足性质$(gH)$,则$T$满足广义Weyl定理.

证  设$T\in L(X)$是polaroid算子,即$iso\sigma(T)\subseteq\Pi(T)$,则$E(T)\subseteq\Pi(T)$.由于$\Pi(T)\subseteq E(T)$对任意的算子$T$都成立,则$E(T)=\Pi(T)$.

$(1)$若$T$满足性质$(H)$,由定理2.7知$T$满足广义Browder定理,则$\sigma(T)\setminus\sigma_{bw}(T)=\Pi(T)=E(T)$,即$T$满足广义Weyl定理.

$(2)$若$T$满足性质$(gH)$,由定理2.6知$T$满足广义Browder定理,同$(1)$的证明,可得$T$满足广义Weyl定理.

设$T\in L(X)$,若$iso\sigma_a(T)\subseteq\Pi^0(T)$,则称$T$是finite-a-polaroid算子.下面定理给出满足性质$(H)$和性质$(gH)$的特殊算子.

定理2.10  设$T$是finite-a-polaroid算子,若$T$或$T^*$有SVEP,则$T$满足性质$(H)$.进而, $T$满足性质$(gH)$.

证  若$T$或$T^*$有SVEP,则$T$满足Browder定理,由定理2.7知,要证$T$满足性质$(H)$,只需证明: $\Pi(T)=E^0_a(T)$.对任意的$\lambda\in\Pi(T)$,有$\lambda\in iso\sigma(T)\subseteq iso\sigma_a(T)$.由于$T$是finite-a-polaroid算子,则$\lambda\in\Pi^0(T)\subseteq E^0(T)\subseteq E^0_a(T)$,即$\Pi(T)\subseteq E^0_a(T)$.反之,对任意的$\lambda\in E^0_a(T)$,则$\lambda\in iso\sigma_a(T)$.由于$T$是finite-a-polaroid算子,则$\lambda\in\Pi^0(T)\subseteq\Pi(T)$,即$E^0_a(T)\subseteq\Pi(T)$.因此, $T$满足性质$(H)$.进而,由定理2.1知$T$满足性质$(gH)$.

下面定理中, $H_{nc}(\sigma(T))$表示在$\sigma(T)$的开邻域的每个连通分支上都不是常数的解析函数空间.

定理2.11  设$T\in L(X)$,若$\sigma_p(T)=\emptyset$,则对$\forall f\in H_{nc}(\sigma(T))$, $f(T)$满足性质$(H)$.进而, $f(T)$满足性质$(gH)$.

证  设$\sigma_p(T)=\emptyset$,由文献[13]的定理2.5知$\sigma_p(f(T))=\emptyset$,则$f(T)$有SVEP,特别地, $f(T)$在$\lambda\notin\sigma_w(T)$处有SVEP.由文献[1]的定理4.23知$f(T)$满足Browder定理,又由定理2.7知,要证$f(T)$满足性质$(H)$,只需证明: $\Pi(f(T))=E^0_a(f(T))$.

由于$\sigma_p(f(T))=\emptyset$,即对$\forall\lambda\in{\Bbb C}$, $\alpha(\lambda I-f(T))=0$,则$E^0_a(f(T))=E(f(T))=\emptyset$,因$\Pi(f(T))\subseteq E(f(T))$,则$\Pi(f(T))=\emptyset=E^0_a(f(T))$.因此, $f(T)$满足性质$(H)$.进而,由定理2.1知$T$满足性质$(gH)$.

设$T\in L(X), $定义

$\begin{eqnarray*}K(T)&=&\Big\{x\in X: \mbox{存在序列$ \{x_n\}_{n\geq1}\subseteq X$和常数$\delta>0$,使得$Tx_1=x, Tx_{n+1}=x_n, $} \\&&\mbox{且$\|x_n\|\leq\delta^n\|x\|, $对$\forall n\geq 1\Big\}$}\end{eqnarray*}$

为$T$的解析核.

推论2.2  设$T\in L(X)$,若$\exists\lambda_0\in{\Bbb C}$,使$K(\lambda_0I-T)=\{0\}$且$N(\lambda_0I-T)=\{0\}$,则对$\forall f\in H_{nc}(\sigma(T))$, $f(T)$满足性质$(H)$.进而, $f(T)$满足性质$(gH)$.

证  容易证明,对$\forall\lambda\neq\lambda_0$,有$(\lambda_0-T)(N(\lambda I-T))=N(\lambda I-T)$,从而由文献[1]的定理1.29可知

$N(\lambda I-T)\subseteq K(\lambda_0I-T), \ \forall\lambda\neq\lambda_0.$

所以对$\forall\lambda\in{\Bbb C}, N(\lambda I-T)=\{0\}$,即$\sigma_p(T)=\emptyset$.由定理2.11知$f(T)$满足性质$(H)$.进而, $f(T)$满足性质$(gH)$.

本节研究了性质$(H)$和性质$(gH)$在可交换的幂零算子、拟幂零算子、有限秩算子和Riesz算子摄动下的稳定性.

定理3.1  设$T\in L(X)$, $N$是与$T$可交换的幂零算子,那么

$(1)$若$T$满足性质$(H)$,则$T+N$满足性质$(H)$;

$(2)$若$T$满足性质$(gH)$,则$T+N$满足性质$(gH)$.

证  设$N$是与$T$可交换的幂零算子,则$\sigma(T)=\sigma(T+N)$,由文献[14]的结论知

$\sigma_{bw}(T)=\sigma_{bw}(T+N), E^0_a(T)=E^0_a(T+N), \Pi^0_a(T)=\Pi^0_a(T+N).$

所以, $(1)(2)$成立.

下面例3.1和3.2分别表明,性质$(H)$和$(gH)$在可交换拟幂零算子和可交换有限秩算子的摄动下都是不稳定的.

例3.1  设$S\in L(l^2({\Bbb N}))$是单边右移位算子, $U\in L(l^2({\Bbb N}))$是拟幂零算子,即

$U(x_1, x_2, \cdots)=(0, x_1, 0, \frac{x_3}{3}, \frac{x_4}{4}, \cdots), \forall(x_n)\in l^2({\Bbb N}), $

设$V\in L(l^2({\Bbb N}))$是拟幂零算子,即

$V(x_1, x_2, \cdots)=(0, 0, 0, -\frac{x_3}{3}, -\frac{x_4}{4}, \cdots), \forall(x_n)\in l^2({\Bbb N}).$

显然, $UV=VU$.考虑算子$T=S\oplus U$和$Q=0\oplus V$,由文献[15]的例$2.14$知, $Q$是拟幂零算子且$TQ=QT$.并且

$\sigma(T)={\Bbb D}(0, 1), \sigma_a(T)={\Bbb C}(0, 1)\cup\{0\}, \sigma(T+Q)={\Bbb D}(0, 1), \sigma_a(T+Q)={\Bbb C}(0, 1)\cup\{0\}, $

以及

$E^0_a(T)=\Pi(T)=\Pi^0_a(T)=\emptyset, \{0\}=E^0_a(T+Q)=\Pi^0_a(T+Q)\neq\Pi(T+Q)=\emptyset, $

其中${\Bbb D}(0, 1)$是${\Bbb C}$中的单位闭圆盘, ${\Bbb C}(0, 1)$是${\Bbb C}$中的单位圆周.由定理$2.7$和$2.8$知$T+Q$不满足性质$(H)$和性质$(gH)$.又因$T$有SVEP,则$T$满足Browder定理,由定理$2.7$和$2.8$知$T$满足性质$(H)$和性质$(gH)$.

例3.2  设$S\in L(l^2({\Bbb N}))$是单边右移位算子.对给定的$0<\varepsilon<1$,设$F_\varepsilon\in L(l^2({\Bbb N}))$定义为: $F_\varepsilon(x_1, x_2, \cdots)=(-\varepsilon x_1, 0, 0, \cdots), (x_n)\in l^2({\Bbb N})$.考虑$T=S\oplus I$和$F=0\oplus F_\varepsilon$,可知$F$是有限秩算子且$FT=TF$.由文献[15]的例$2.12$知

$\sigma(T)={\Bbb D}(0, 1), \sigma_a(T)={\Bbb C}(0, 1), \sigma(T+F)={\Bbb D}(0, 1), \sigma_a(T+F)={\Bbb C}(0, 1)\cup\{0, 1-\varepsilon\}, $

且

$E^0_a(T)=\Pi(T)=\Pi^0_a(T)=\emptyset, \{1-\varepsilon\}=E^0_a(T+F)=\Pi^0_a(T+F)\neq\Pi(T+F)=\emptyset, $

其中${\Bbb D}(0, 1)$是${\Bbb C}$中的单位闭圆盘, ${\Bbb C}(0, 1)$是${\Bbb C}$中的单位圆周.由定理$2.7$和$2.8$知$T+F$不满足性质$(H)$和性质$(gH)$.又因$T$有SVEP,则$T$满足Browder定理,由定理$2.7$和$2.8$知$T$满足性质$(H)$和性质$(gH)$.

由于拟幂零算子和有限秩算子都是特殊的Riesz算子,上述例子表明性质$(H)$和性质$(gH)$在可交换Riesz算子的摄动下也是不稳定的.下面定理给出它们在可交换Riesz算子摄动下稳定的条件.

定理3.2  设$T\in L(X)$, $R$是与$T$可交换的Riesz算子,那么

$(1)$若$T$满足性质$(H)$,则$T+R$满足性质$(H)\Leftrightarrow\Pi(T+R)=E^0_a(T+R)$;

$(2)$若$T$满足性质$(gH)$,则$T+R$满足性质$(gH)\Leftrightarrow\Pi(T+R)=\Pi^0_a(T+R)$.

证  $(1)$设$T+R$满足性质$(H)$,由定理2.7知$\Pi(T+R)=E^0_a(T+R)$.反之,设$\Pi(T+R)=E^0_a(T+R)$,因$T$满足性质$(H)$,由定理2.7知$T$满足Browder定理.又由文献[1]的定理4.29知$T+R$满足Browder定理.故由定理2.7知$T+R$满足性质$(H)$.同理由定理2.8可证$(2)$成立.

设$T\in L(X)$,若$iso\sigma_a(T)\subseteq E_a^0(T)$,则称$T$是finite-a-isoloid算子.于是,有下面定理.

定理3.3  设$T\in L(X)$是finite-a-isoloid算子, $R$是与$T$可交换的Riesz算子, $\sigma_a(T+R)=\sigma_a(T)$.若$T$满足性质$(H)$,则$T+R$也满足性质$(H)$.

证  设$T$满足性质$(H)$,由定理3.2知,只需证明: $\Pi(T+R)=E^0_a(T+R).$

对任意的$\lambda\in\Pi(T+R)$,有$\lambda\in iso\sigma(T+R)\subseteq iso\sigma_a(T+R)$,由于$\sigma_a(T+R)=\sigma_a(T)$,则$\lambda\in iso\sigma_a(T)$.因$T$是finite-a-isoloid算子,则$\lambda\in E^0_a(T)$.又因$T$满足性质$(H)$,由定理2.3知$\lambda\in E^0_a(T)=\Pi^0_a(T)=\sigma_a(T)\setminus\sigma_{ub}(T)$,即$\lambda I-T$是上半Browder算子,由文献[1]的推论2.81知, $\lambda I-(T+R)$是上半Browder算子,则

$\lambda\in\sigma_a(T+R)\setminus\sigma_{ub}(T+R)=\Pi^0_a(T+R)\subseteq E^0_a(T+R).$

所以, $\Pi(T+R)\subseteq E^0_a(T+R)$.

反之,对任意的$\lambda\in E^0_a(T+R)$,有$\lambda\in iso\sigma_a(T+R)$,同上可证$\lambda\in E^0_a(T)$.又因$T$满足性质$(H)$,由定理2.3知$\lambda\in E^0_a(T)=\Pi^0(T)=\sigma(T)\setminus\sigma_b(T)$,即$\lambda I-T$是Browder算子,由文献[1]的推论2.81知, $\lambda I-(T+R)$是Browder算子,即

$\lambda\in\sigma(T+R)\setminus\sigma_b(T+R)=\Pi^0(T+R)\subseteq\Pi(T+R), $

即$E^0_a(T+R)\subseteq\Pi(T+R)$.因此, $T+R$也满足性质$(H)$.

推论3.1  设$T\in L(X)$是finite-a-isoloid算子, $Q$是与$T$可交换的拟幂零算子,若$T$满足性质$(H)$,则$T+Q$也满足性质$(H)$.

推论3.2  设$T\in L(X)$是finite-a-isoloid算子, $F$是与$T$可交换的有限秩算子, $\sigma_a(T+F)=\sigma_a(T)$.若$T$满足性质$(H)$,则$T+F$也满足性质$(H)$.

类比定理3.3的证明过程,利用定理2.2、2.8和3.2同理可证,满足某些条件时性质$(gH)$在可交换Riesz算子的摄动下也具有稳定性,即定理3.4成立.

定理3.4  设$T\in L(X)$是a-polaroid算子, $R$是与$T$可交换的Riesz算子, $\sigma_a(T+R)=\sigma_a(T)$.若$T$满足性质$(gH)$,则$T+R$也满足性质$(gH)$.

定理3.5  设$T\in L(X)$是polaroid算子, $R$是与$T$可交换的Riesz算子, $\sigma_a(T+R)=\sigma_a(T), $$ \sigma(T +R)=\sigma(T)$.若$T$满足性质$(gH)$,则$T+R$也满足性质$(gH)$.

证  设$T$满足性质$(gH)$,由定理3.2知,只需证明: $\Pi(T+R)=\Pi^0_a(T+R)$.

对任意的$\lambda\in\Pi(T+R)$,有$\lambda\in iso\sigma(T+R)\subseteq iso\sigma_a(T+R)$,即有$\lambda\in\sigma_a(T+R)$.由于$\sigma(T+R)=\sigma(T)$,则$\lambda\in iso\sigma(T)$.因$T$是polaroid算子,则$\lambda\in\Pi(T)$.又因$T$满足性质$(gH)$,由定理2.8知$\lambda\in\Pi(T)=\Pi^0_a(T)=\sigma_a(T)\setminus\sigma_{ub}(T)$,即$\lambda I-T$是上半Browder算子,则$\lambda I-(T+R)$也是上半Browder算子,所以$\lambda\in\sigma_a(T+R)\setminus\sigma_{ub}(T+R)=\Pi^0_a(T+R)$,即$\Pi(T+R)\subseteq\Pi^0_a(T+R)$.

反之,对任意的$\lambda\in\Pi^0_a(T+R)=\sigma_a(T+R)\setminus\sigma_{ub}(T+R)$,有$\lambda I-(T+R)$是上半Browder算子,则$\lambda I-T$也是上半Browder算子,即$\lambda\notin\sigma_{ub}(T)$.由于$\lambda\in iso\sigma_a(T+R)$且$\sigma_a(T+R)=\sigma_a(T)$,则$\lambda\in iso\sigma_a(T)$,从而$\lambda\in\sigma_a(T)\setminus\sigma_{ub}(T)=\Pi^0_a(T)$.因$T$满足性质$(gH)$，由定理2.2知$\lambda\in\Pi^0_a(T)=\Pi^0(T)=\sigma(T)\setminus\sigma_b(T)$,即$\lambda I-T$是Browder算子,则$\lambda I-(T+R)$也是Browder算子,所以$\lambda\in\sigma(T+R)\setminus\sigma_b(T+R)=\Pi^0(T+R)\subseteq\Pi(T+R)$,即$\Pi^0_a(T+R)\subseteq\Pi(T+R)$.因此, $T+R$也满足性质$(gH)$.

推论3.3  设$T\in L(X)$是polaroid算子, $Q$是与$T$可交换的拟幂零算子,若$T$满足性质$(gH)$,则$T+Q$也满足性质$(gH)$.

定理3.6  设$T\in L(X)$是polaroid算子, $F$是与$T$可交换的有限秩算子, $\sigma_a(T+F)=\sigma_a(T)$.若$T$满足性质$(gH)$,则$T+F$也满足性质$(gH)$.

证  因有限秩算子是Riesz算子,由定理3.2知,只需证明: $\Pi(T+F)=\Pi^0_a(T+F)$.对任意的$\lambda\in\Pi(T+F)$,有$\lambda\in iso\sigma(T+F)\subseteq iso\sigma_a(T+F)$.因$F$是与$T$可交换的有限秩算子,由文献[16]的定理2.2可知, $acc\sigma(T)=acc\sigma(T+F)$,则$\lambda\in iso\sigma(T)$或$\lambda\notin\sigma(T)$.若$\lambda\in iso\sigma(T)$,因$T$是polaroid算子,则$\lambda\in\Pi(T)$.又因$T$满足性质$(gH)$,由定理2.8知$\lambda\in\Pi(T)=\Pi^0_a(T)=\sigma_a(T)\setminus\sigma_{ub}(T)$,即$\lambda I-T$是上半Browder算子,则$\lambda I-(T+F)$是上半Browder算子,所以$\lambda\in\sigma_a(T+F)\setminus\sigma_{ub}(T+F)=\Pi^0_a(T+F)$.若$\lambda\notin\sigma(T)$,显然$\lambda I-T$是上半Browder算子,同理可证$\lambda\in\Pi^0_a(T+F)$.因此, $\Pi(T+F)\subseteq\Pi^0_a(T+F)$.

反之,对任意的$\lambda\in\Pi^0_a(T+F)$,有$\lambda\in iso\sigma_a(T+F)$且$\lambda I-(T+F)$是上半Browder算子,则$\lambda I-T$也是上半Browder算子,即$\lambda\notin\sigma_{ub}(T)$.由于$\sigma_a(T+F)=\sigma_a(T)$,则$\lambda\in\sigma_a(T)\setminus\sigma_{ub}(T)=\Pi^0_a(T)$.又因$T$满足性质$(gH)$,由定理2.8知$\lambda\in\Pi^0_a(T)=\Pi(T)=\sigma(T)\setminus\sigma_d(T)$.因$F$是与$T$可交换的有限秩算子,由文献[14]的定理2.11知, $\sigma_d(T)=\sigma_d(T+F)$,则$\lambda\in\sigma(T+F)\setminus\sigma_d(T+F)=\Pi(T+F)$.所以, $\Pi^0_a(T+F)\subseteq\Pi(T+F)$.因此, $T+F$也满足性质$(gH)$.

此外,性质$(H)$和$(gH)$在可交换有限秩算子的摄动下还有下面结论.

定理3.7  设$T\in L(X)$, $F$是与$T$可交换的有限秩算子,如果$iso\sigma_a(T)=\emptyset$,那么

$(1)$若$T$满足性质$(H)$,则$T+F$满足性质$(H)$;

$(2)$若$T$满足性质$(gH)$,则$T+F$满足性质$(gH)$.

证  $(1)$若$iso\sigma_a(T)=\emptyset$,则$iso\sigma_a(T)\subseteq E^0_a(T)$,即$T$是finite-a-isoloid算子.因$F$是与$T$可交换的有限秩算子且$iso\sigma_a(T)=\emptyset$,由文献[17]的引理2.6可知, $\sigma_a(T+F)=\sigma_a(T)$,所以由推论3.2知$(1)$成立.

$(2)$若$iso\sigma_a(T)=\emptyset$,则$iso\sigma(T)\subseteq iso\sigma_a(T)\subseteq\Pi(T)$,即$T$是polaroid算子.因$F$是与$T$可交换的有限秩算子且$iso\sigma_a(T)=\emptyset$,由文献[17]的引理2.6可知, $\sigma_a(T+F)=\sigma_a(T)$,所以由定理3.6可知, $(2)$成立.

定理3.8  设$T\in L(X)$是单射拟幂零算子, $F$是与$T$可交换的有限秩算子,那么

$(1)$若$T$满足性质$(H)$,则$T+F$满足性质$(H)$;

$(2)$若$T$满足性质$(H)$,则$T+F$满足性质$(H)$.

证  设$T$是单射拟幂零算子,因$TF$是有限秩算子,则$TF$是幂零算子.因$T$是单射的,则$F$是幂零算子,由定理3.1知$(1)(2)$成立.