Property and Property for Operator and Its Functions

Chenhui SUN; Xiaohong CAO

doi:10.1051/wujns/2025301060

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Volume 30 / No 1 (February 2025)

Wuhan Univ. J. Nat. Sci., 30 1 (2025) 60-68

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Issue		Wuhan Univ. J. Nat. Sci. Volume 30, Number 1, February 2025


Page(s)		60 - 68
DOI		https://doi.org/10.1051/wujns/2025301060
Published online		12 March 2025

Wuhan University Journal of Natural Sciences, 2025, Vol.30 No.1, 60-68

Mathematics

CLC number: O177

Property $(W_{E})$ and Property $(R)$ for Operator and Its Functions

算子及算子函数的 $(W_{E})$ 性质与 $(R)$ 性质

Chenhui SUN (孙晨辉)¹ and Xiaohong CAO (曹小红)²

¹ School of Mathematics and Statistics, Weinan Normal University, Weinan 714099, Shaanxi, China
² School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, Shaanxi, China

Received: 18 April 2024

Abstract

By using the topological uniform descent, the necessary and sufficient conditions for which property $(W_{E})$ and property $(R)$ hold for bounded linear operators are given. As a consequence of the main result, the stability of property $(W_{E})$ and property $(R)$ is studied, and a new judgement for operator functions that satisfy property $(W_{E})$ and property $(R)$ is discussed.

摘要

本文运用拓扑一致降标性质，给出了有界线性算子同时满足 $(W_{E})$ 性质与 $(R)$ 性质的充要条件。之后利用主要结论，研究了 $(W_{E})$ 性质与 $(R)$ 性质的稳定性，并得到了算子函数同时有 $(W_{E})$ 性质与 $(R)$ 性质的新判定方法。

Key words: property (W_E) / property (R) / spectrum

关键字 : (W_E)性质 / (R)性质 / 谱

Cite this article: SUN Chenhui, CAO Xiaohong. Property (W_E) and Property (R) for Operator and Its Functions[J]. Wuhan Univ J of Nat Sci, 2025, 30(1): 60-68.

Biography: SUN Chenhui, female, Ph. D., Associate professor, research direction: operator theory. E-mail: sunchenhui1986@163. com

Foundation item: Supported by the 2021 General Special Scientific Research Project of the Education Department of Shaanxi Provincial Government (21JK0637), the Science and Technology Planning Project of Weinan Science and Technology Bureau (2022ZDYFJH-11) and the 2021 Talent Project of Weinan Normal University (2021RC16)

© Wuhan University 2025

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The Weyl type theorem of bounded linear operator can well reflect the structure characteristics and distribution of operator's spectral^[1-5]. Therefore, the Weyl type theorem is an important topic in spectral theory. In recent years, the research scope of Weyl type theorems has been extended from general operators to operator functions, operator matrices, etc. Numerous significant results have been obtained^[6-9]. Property $(W_{E})$ and property $(R)$ are the latest variations of Weyl type theorems, which have attracted the attention and research of operator theorists^[10-12]. In this paper, by decomposing and constructing the operator spectrum and using the topological uniform scaling property, we give a new method for bounded linear operators and operator functions to satisfy the property $(W_{E})$ and property $(R)$ . Furthermore, the perturbation of property $(W_{E})$ and property $(R)$ is characterized, and the necessary and sufficient conditions for the operator function to have both the property $(W_{E})$ and property $(R)$ are studied.

Throughout this paper, $H$ denotes a complex separable infinite dimensional Hilbert space. Let $B (H)$ be the algebra of all bounded linear operators on $H$ . For an operator $T \in B (H)$ we shall denote by $n (T)$ the dimension of the kernel $N (T)$ , and by $d (T)$ the codimension of the range $R (T)$ . We call $T \in B (H)$ is an upper semi-Fredholm operator if $n (T) < \infty$ and $R (T)$ is closed. If $T \in B (H)$ is an upper semi-Fredholm operator and $n (T) = 0$ , we call $T$ a bounded below operator. If $d (T)$ $< \infty$ , $T$ is a lower semi-Fredholm operator. An operator $T \in B (H)$ is said to be Fredholm if $R (T)$ is closed and both $n (T)$ and $d (T)$ are finite. If $T \in B (H)$ is an upper (or a lower) semi-Fredholm operator, the index of $T$ , $i n d (T)$ , is defined to be $i n d (T) =$ $n (T) - d (T)$ . The ascent of $T$ , $a s c (T)$ , is the least non-negative integer $n$ such that $N (T^{n}) =$ $N (T^{n + 1})$ and the descent, $d e s (T)$ , is the least non-negative integer $n$ such that $R (T^{n})$ $=$ $R (T^{n + 1})$ . We call $T$ a Drazin invertible operator if $a s c (T)$ $= d e s (T) < \infty$ . The operator $T$ is Weyl if it is Fredholm of index zero, and $T$ is said to be Browder if it is Fredholm "of finite ascent and descent". Let $σ (T)$ be the spectrum of $T$ and $σ_{a} (T)$ be the approximate point spectrum of $T$ . We write $σ_{w} (T)$ , $σ_{b} (T)$ , $σ_{e} (T)$ , $σ_{S F} (T)$ , $σ_{e a} (T)$ and $σ_{a b} (T)$ for the Weyl spectrum of $T$ , the Browder spectrum of $T$ , the essential spectrum of $T$ , the semi-Fredholm spectrum of $T$ , the essential approximate point spectrum of $T$ and the Browder essential approximate point spectrum of $T$ . Let $ρ (T) =$ $C \ σ (T)$ , $ρ_{a} (T) =$ $C \ σ_{a} (T)$ , $ρ_{w} (T) =$ $C \ σ_{w} (T)$ , $ρ_{b} (T) =$ $C \ σ_{b} (T)$ , $ρ_{e} (T) =$ $C \ σ_{e} (T)$ , $ρ_{a b} (T) =$ $C \ σ_{a b} (T)$ ( $C$ denotes the set of complex numbers). $T \in B (H)$ is called an isoloid operator if $i s o σ (T)$ $\subseteq σ_{p} (T)$ where $σ_{p} (T)$ $= {λ \in C : n (T - λ I) > 0}$ . For a set $X \subseteq C$ , we write $i s o X$ , $a c c X$ and $\partial X$ for the set of isolated points, accumulation points and boundary points set of $X$ . We denote by $σ_{0} (T)$ the set of all normal eigenvalues of $T$ , thus $σ_{0} (T) = σ (T) \ σ_{b} (T)$ . $T \in B (H)$ satisfies property $(W_{E})$ if $σ (T) \ σ_{w} (T) = E (T)$ , where $E (T) = {λ \in i s o σ (T) : n (T - λ I) >$ $0}$ ^[10]. $T \in B (H)$ satisfies property $(R)$ if $σ_{a} (T) \ σ_{a b} (T) = π_{00} (T)$ , where $π_{00} (T) = {λ \in i s o σ (T) : 0 < n (T - λ I) < + \infty}$ ^[10].

In this paper, we mainly study property $(W_{E})$ and property $(R)$ for bounded linear operators and its functions. Some meaningful conclusions are obtained.

1 Judgement of Property $(W_{E})$ and Property $(R)$ for Bounded Linear Operators

Although both property $(W_{E})$ and property $(R)$ are variations of Weyl's theorem based on their definitions, there is no necessary connection between them.

Remark 1 (i) $T \in B (H)$ satisfies property $(W_{E})$ $⇏$ $T$ satisfies property $(R)$ .

Let $A, B \in B (l^{2})$ be defined by $A (x_{1}, x_{2}, x_{3}, \dots) = (0, x_{1}, x_{2}, \dots)$ , $B (x_{1}, x_{2}, x_{3}, \dots) = (0, x_{2}, x_{3}, \dots)$ . Suppose that $T = (\begin{matrix} A & 0 \\ 0 & B \end{matrix})$ . Then $σ (T) = {λ \in C : | λ | \leq 1} = σ_{w} (T)$ , $E (T) = π_{00} (T) = \emptyset$ , $σ_{a} (T) = {λ \in C : | λ | = 1} ⋃ {0}$ , $σ_{a b} (T) = {λ \in C : | λ | = 1}$ . $T$ satisfies property $(W_{E})$ , but property $(R)$ does not hold for $T$ .

(ii) $T \in B (H)$ satisfies property $(R)$ $⇏$ $T$ satisfies property $(W_{E})$ .

Let $A, B \in B (l^{2})$ be defined by $A (x_{1}, x_{2}, x_{3}, \dots) = (0, x_{1}, x_{2}, \dots)$ , $B (x_{1}, x_{2}, x_{3}, \dots) = (x_{2}, x_{3}, \dots)$ , $T = (\begin{matrix} A & 0 \\ 0 & B \end{matrix})$ . We have that $σ (T) = {λ \in C : | λ | \leq 1} \neq σ_{w} (T)$ , $σ_{a} (T) = σ_{a b} (T) = {λ \in C : | λ | \leq 1}$ . $π_{00} (T) = E (T) = \emptyset$ . So $T$ satisfies property $(R)$ and $T$ does not have property $(W_{E})$ .

(iii) $T \in B (H)$ has property $(R)$ and property $(W_{E})$ $\Leftrightarrow$ $σ_{w} (T) = σ_{a b} (T) ⋃ [ρ_{a} (T) ⋂ σ (T)]$ and $π_{00} (T) = E (T) = σ_{0} (T)$ .

Topological uniform descent is an important property of operators, which is widely used in spectral theory. If $T \in B (H)$ , then for each nonnegative integer $n$ , $T$ induces a linear transformation from the vector space $R (T^{n}) / R (T^{n + 1})$ to $R (T^{n + 1}) / R (T^{n + 2})$ . We denote $k_{n} (T)$ the dimension of the null space of the induced map and let $k (T) = \sum_{n = 0}^{\infty} k_{n} (T)$ . The following definition was introduced by Grabiner^[13]. Let $T \in B (H)$ , if there is a nonnegative integer $d$ for which $k_{n} (T) = 0$ for $n \geq d$ and $R (T^{n})$ is closed in the operator range topology of $R (T^{d})$ for $n \geq d$ , then we say that $T$ has topological uniform descent.

It can be shown that if $T$ is upper semi-Fredholm, then $T$ has topological uniform descent. Let $ρ_{τ} (T) = {λ \in C : T - λ I$ has topological uniform descent}, $σ_{τ} (T) = C \ ρ_{τ} (T)$ . Grabiner discovered many properties of topological uniform descent. We will use the following property (Ref. [13], Corollary 4.9): Suppose that $T \in B (H)$ , $λ \in \partial σ (T)$ , if $T - λ I$ has topological uniform descent, then $λ$ is a pole of $T$ . Next, we will discuss property $(W_{E})$ and property $(R)$ by using the topological uniform descent.

Lemma 1 Let $T \in B (H)$ , the following statements are equivalent:

(1) $T$ satisfies property $(W_{E})$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0}$ ;

(3) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{e a} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)} .$

Proof (1) $\Rightarrow$ (2). Suppose that $T$ satisfies property $(W_{E})$ . If $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T) ⋃$ ${λ \in σ (T) : n (T - λ I) = 0}$ , then there exists a deleted neighborhood $B^{°} (λ_{0})$ centered on $λ_{0}$ such that for any $λ \in B^{°} (λ_{0})$ , $T - λ I$ is a Weyl operator. Since $T$ has property $(W_{E})$ , we know that $T - λ I$ is a Browder operator. Then we get that $λ_{0} \in ρ (T) \cup \partial σ (T)$ . We assume that $λ_{0} \in \partial σ (T)$ . If $λ_{0} \in ρ_{τ} (T)$ , we get that $T - λ_{0} I$ is Drazin invertible. According to $n (T - λ_{0} I) > 0$ , we have $λ_{0} \in E (T)$ . Since $T$ satisfies property $(W_{E})$ , we can get that $T - λ_{0} I$ is a Browder operator. If $λ_{0} \notin a c c σ (T)$ , we can also get that $λ_{0} \in E (T)$ and $T - λ_{0} I$ is Browder operators. The inclusion " $\supseteq$ " is obviously true.

(2) $\Rightarrow$ (1). Since $σ (T) \ σ_{w} (T)] ⋂ [σ_{τ} (T) ⋂ a c c σ_{a} (T)] = \emptyset$ and $[σ (T) \ σ_{w} (T)] ⋂ [a c c σ_{w} (T) ⋃$ ${λ \in σ (T) : n (T - λ I) = 0}] = \emptyset$ , we know that $σ (T) \ σ_{w} (T) \subseteq E (T)$ . Similarly, $E (T) \subseteq$ $σ (T) \ σ_{w} (T)$ . Hence $T$ satisfies property $(W_{E})$ .

The fact that $a c c σ_{e a} (T) ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)} = a c c σ_{w} (T)$ implies that (2) $\Leftrightarrow$ (3).

The closeness of operator range is very important in spectral theory. According to the closeness of operator range, the following spectral set is defined: $σ_{c} (T) = {λ \in C : R (T - λ I)$ is not closed}.

Lemma 2 Let $T \in B (H)$ , the following statements are equivalent:

(1) $T$ satisfies property $(R)$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T$

$d (T - λ I)} ⋂ σ_{c} (T)]$ ;

(3) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < - λ$

$I)} ⋂ σ_{c} (T)]$

Proof (1) $\Rightarrow$ (2). Suppose $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ$

$\in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Without loss of generality, we assume that $λ_{0} \in σ (T)$ . Then we have $0 < n (T - λ_{0} I) < \infty$ . There exists a deleted neighborhood $B^{°} (λ_{0}; ε)$ centered on $λ_{0}$ such that for any $λ \in B^{°} (λ_{0}; ε)$ , $λ \in ρ_{a b} (T)$ . If $λ_{0} \notin a c c σ (T)$ , then $λ_{0} \in π_{00} (T)$ . Since $T$ satisfies property $(R)$ , we have that $T - λ_{0} I$ is a Browder operator. Suppose that $λ_{0} \in ρ_{τ} (T)$ , if $λ_{0} \in ρ_{c} (T)$ , then $T - λ_{0} I$ is an upper semi-Fredholm operator. By $\forall λ \in B^{°} (λ_{0}; ε)$ , $λ \in ρ_{a b} (T)$ and $λ_{0} \in i s o σ_{a} (T)$ , we know that $λ_{0} \in σ_{a} (T) \ σ_{a b} (T)$ . Because $T$ has property $(R)$ , we can get that $λ_{0} \in ρ_{b} (T)$ . If $λ_{0} \notin a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ , then exists a deleted neighborhood $B^{°} (λ_{0}) \subseteq B^{°} (λ_{0}; ε)$ centered on $λ_{0}$ such that for any $λ \in B^{°} (λ_{0})$ , $λ \in ρ_{b} (T)$ . Therefore $λ_{0} \in \partial σ (T)$ . Since $λ_{0} \in ρ_{τ} (T)$ and $n (T - λ_{0} I) < \infty$ , we can also get that $λ_{0} \in ρ_{b} (T)$ .

(2) $\Rightarrow$ (1). Suppose that $λ_{0} \in σ_{a} (T) \ σ_{a b} (T)$ . It follows that $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) <$ $d (T - λ I)} ⋂ σ_{c} (T)]$ . So $T - λ_{0} I$ is a Browder operator. Then we have $σ_{a} (T) \ σ_{a b} (T) \subseteq π_{00} (T)$ . Similarly, we can get the inclusion " $\supseteq$ ". Hence $T$ satisfies property $(R)$ .

If (2) holds, then $T$ has property $(R)$ . This implies that ${λ \in C : 0 < n (T - λ I) < \infty} ⋂ ρ_{c} (T) ⋂ a c c [ρ_{a} (T) ⋂ σ (T)] = \emptyset$ , where $ρ_{c} (T) = C \ σ_{c} (T)$ . It follows that $a c c σ (T) = a c c σ_{a} (T) ⋃ {a c c [ρ_{a} (T) ⋂ σ (T)] ⋂ σ_{c} (T)} ⋃ {a c c [ρ_{a} (T) ⋂ σ (T)] ⋂ ρ_{c} (T) ⋂ {λ$

$\in σ (T) : n (T - λ I) = 0}} ⋃ {a c c [ρ_{a} (T) ⋂ σ (T)] ⋂ ρ_{c} (T) ⋂ {λ \in C : n (T - λ I) = \infty}}$ . Hence $[σ_{τ} (T) ⋂ a c c σ (T)] \subseteq {[σ_{τ} (T) ⋂ a c c σ_{a} (T)]$

$⋃$ ${λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C :$ $n (T - λ I) < d (T -$ $λ I)} ⋂ σ_{c} (T)]}$ . Then we have that $σ_{b} (T) \subseteq [σ_{τ}$

$(T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$

The inclusion " $\supseteq$ " is obviously true. This implies that (3) holds. For the converse, if (3) holds, by $a c c σ_{a} (T) \subseteq a c c σ (T)$ we know that (2) holds. Therefore (2) $\Leftrightarrow$ (3).

Based on Lemma 1 and Lemma 2, the following results demonstrate that the two properties can be valid at the same time:

Theorem 1 $T \in B (H)$ satisfies property $(R)$ and property $(W_{E})$ if and only if $\begin{array}{l} σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} \\ (T) ⋂ σ_{w} (T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] . \end{array}$

Proof " $\Rightarrow$ ". Suppose $T$ has property $(R)$ . Using Lemma 2, we have that $\begin{array}{l} σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{a b} (T) ⋃ \\ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] . \end{array}$ Since $T$ has property $(W_{E})$ , we know that $ρ_{w} (T) = ρ_{b} (T) a n d E (T) = σ_{0} (T) . H e n c e a c c σ_{a b} (T) ⋂ ρ_{w} (T) = \emptyset, {λ \in i s o σ (T) : n (T - λ I) = \infty} = \emptyset .$ Thus $\begin{array}{l} a c c σ_{a b} (T) = [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ [a c c σ_{a b} (T) ⋂ ρ_{w} (T)] = a c c σ_{a b} (T) ⋂ σ_{w} (T) . {λ \in C : n (T - λ I) = \infty} = {λ \in a c c σ (T) : n (T - λ I) = \infty} \\ ⋃ {λ \in i s o σ (T) : n (T - λ I) = \infty} = {λ \in a c c σ (T) : n (T - λ I) = \infty} . T h e r e f o r e σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ \\ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] . \end{array}$

" $\Leftarrow$ ". By the condition, we get that $σ_{b} (T) \subseteq [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] .$ The inclusion " $\supseteq$ " is obviously true. From Lemma 2, we know that $T$ satisfies property $(R)$ .

If $λ_{0} \in σ (T) \ σ_{w} (T)$ , then $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] . T h u s T - λ_{0} I$ is a Browder operator, $σ (T) \ σ_{w} (T) \subseteq E (T)$ . The converse is similar. Therefore $T$ satisfies property $(W_{E})$ .

Remark 2 (i) In Theorem 1, suppose $T \in B (H)$ satisfies property $(R)$ and property $(W_{E})$ , then each part of the decomposition of $σ_{b} (T)$ cannot be deleted.

( $a$ ) Let $T \in B (l^{2})$ be defined by $T (x_{1}, x_{2}, x_{3}, \dots) = (0, \frac{x_{2}}{2}, \frac{x_{3}}{3}, \dots)$ . Hence $T$ has property $(R)$ and property $(W_{E})$ . But $σ_{b} (T) \neq [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] .$ Thus $[σ_{τ} (T) ⋂ a c c σ_{a} (T)]$ cannot be deleted.

( $b$ ) Let $T \in B (l^{2})$ be defined by $T (x_{1}, x_{2}, x_{3}, \dots) = (x_{2}, x_{3}, x_{4}, \dots)$ . Then $T$ has property $(R)$ and property $(W_{E})$ . But $σ_{b} (T) \neq [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0}$ $⋃ {λ \in a c c σ (T) :$ $n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Thus $[a c c σ_{a b} (T) ⋂ σ_{w} (T)]$ cannot be deleted.

( $c$ ) Let $T \in B (l^{2})$ be defined by $T (x_{1}, x_{2}, x_{3}, \dots) = (0, x_{1}, x_{2}, x_{3}, \dots)$ . We know that $T$ has property $(R)$ and property $(W_{E})$ . But $σ_{b} (T) \neq [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in a c c σ (T) :$ $n (T - λ I)$ $= \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Hence ${λ \in σ (T) : n (T - λ I) = 0}$ cannot be deleted.

( $d$ ) Let $A, B \in B (l^{2})$ be defined by $A (x_{1}, x_{2}, \dots) = (0, x_{1}, x_{2}, \dots)$ , $B (x_{1}, x_{2}, x_{3}, \dots) = (x_{1}, 0, x_{3}, 0, \dots)$ . And suppose $T = (\begin{matrix} A & 0 \\ 0 & B \end{matrix})$ . Then $T$ satisfies property $(R)$ and property $(W_{E})$ . But $σ_{b} (T) \neq [σ_{τ} (T) ⋂$ $a c c σ_{a} (T)] ⋃$ $[a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ [a c c {λ \in C : n (T - λ I) <$ $d (T - λ I)}$ $⋂ σ_{c} (T)]$ . Therefore ${λ \in a c c σ (T) : n (T - λ I) = \infty}$ cannot be deleted.

( $e$ ) Let $A, B \in B (l^{2})$ be defined by $A (x_{1}, x_{2}, \dots) = (0, x_{1}, x_{2}, \dots)$ , $B (x_{1}, x_{2}, x_{3}, \dots) =$ $(0,0, \frac{x_{2}}{2}, \frac{x_{3}}{3}, \dots)$ . And suppose $T = (\begin{matrix} A & 0 \\ 0 & B \end{matrix})$ . Thus $T$ satisfies property $(R)$ and property $(W_{E})$ . But $σ_{b} (T) \neq [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂$ $σ_{w} (T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃$ ${λ \in$ $a c c σ (T) :$ $n (T - λ I) = \infty}$ . It follows that $[a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ cannot be deleted.

(ii) The conditions in Theorem 1 can be transformed as follows: $a c c σ_{a} (T)$ can be replaced by $a c c σ (T)$ , $a c c σ_{a b} (T) ⋂ σ_{w} (T)$ can be replaced by $a c c σ_{a b} (T) ⋂ a c c σ_{w} (T)$ .

From Lemma 1 and Lemma 2, we can get the following Corollary.

Corollary 1 Let $T \in B (H)$ . Then:

(1) $T$ satisfies property $(W_{E})$ $\Leftrightarrow$ $E (T) \subseteq ρ_{τ} (T) \subseteq ρ_{b} (T) ⋃ a c c σ_{w} (T)$ $\Leftrightarrow$ $E (T) \subseteq ρ_{τ} (T) \subseteq$ $ρ_{b} (T) ⋃ a c c σ_{e a} (T) ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ ;

(2) $T$ satisfies property $\begin{array}{l} (R) \Leftrightarrow π_{00} (T) \subseteq ρ_{τ} (T) \subseteq ρ_{b} (T) ⋃ a c c σ_{a} (T) ⋃ {λ \in C : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c \\ {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] \Leftrightarrow π_{00} (T) \subseteq ρ_{τ} (T) \subseteq ρ_{b} (T) ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - [λ I)} ⋂ σ c (T)]] . \end{array}$

By Theorem 1 and Corollary 1, we can get the following Corollary.

Corollary 2 Let $T \in B (H)$ . Then $T$ satisfies property $(R)$ and property $(W_{E}) \Leftrightarrow E (T) \subseteq ρ_{τ} (T) \subseteq ρ_{b} (T) ⋃ [a c c σ_{a b} (T) ⋂$

$σ_{w} (T)] ⋃ {λ \in C : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] .$

Proof Using Theorem 1, $T$ satisfies property $(R)$ and property $(W_{E}) \Leftrightarrow σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w}$

$(T)] ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] .$

" $\Rightarrow$ ". $E (T) \subseteq ρ_{τ} (T)$ is obvious. If $λ_{0} \in ρ_{τ} (T)$ and $λ_{0} \notin [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in σ (T) :$ $n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ , by Theorem 1 we know that $λ_{0} \in ρ_{b} (T)$ .

" $\Leftarrow$ ". Suppose $λ_{0} \in σ (T)$ and $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in σ (T) :$ $n (T - λ I) = 0} ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . If $λ_{0} \in ρ_{τ} (T)$ , we have $λ_{0} \in ρ_{b} (T)$ by the condition. Suppose $λ_{0} \notin a c c σ_{a} (T)$ , if $λ_{0} \in ρ_{c} (T)$ , we can get that $λ_{0} \in ρ_{τ} (T)$ , then $T - λ_{0} I$ is a Browder operator. If $λ_{0} \notin a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ , using $λ_{0} \notin a c c σ_{a} (T)$ we have $λ_{0} \in i s o σ (T)$ . It also follows that $λ_{0} \in ρ_{b} (T)$ . From Theorem 1, we can get that $T$ satisfies property $(R)$ and property $(W_{E})$ .

2 The Perturbation of Property $(W_{E})$ and Property $(R)$

Suppose $T \in B (H)$ satisfies property $(R)$ or property $(W_{E})$ and $K \in B (H)$ is a compact operator or even a finite rank operator, but we cannot deduce that $T + K$ satisfies property $(R)$ or property $(W_{E})$ . For example, let $T, K \in B (l^{2})$ be defined by $T (x_{1}, x_{2}, x_{3}, \dots) = (0, x_{1}, \frac{x_{2}}{2}, \frac{x_{3}}{3}, \dots)$ , $K (x_{1}, x_{2}, x_{3}, \dots) =$ $(0, x_{1}, 0,0, \dots)$ . It is obviously that $T$ satisfies property $(R)$ and property $(W_{E})$ and $K$ is a finite rank operator. However, calculations indicate that $T + K$ does not satisfy either property $(R)$ or property $(W_{E})$ .

The operator $F$ is called a power finite rank operator, if there is a positive integer $n$ such that $d i m R (F^{n}) < \infty$ . A power finite rank operator is a Riesz operator, therefore, when $F$ is a power finite rank operator, operator $T$ is commutative with $F$ , then it has the following properties: $σ_{1} (T + F) = σ_{1} (T)$ , where $σ_{1} \in {σ_{w}, σ_{b},$ $σ_{e}, σ_{S F}, σ_{e a}}$ and $i s o σ (T + F) \subseteq i s o σ (T) ⋃ ρ (T)$ ^[14].

Due to the complexity of general compact perturbations, this section focuses on the power finite rank perturbation of property $(W_{E})$ and property $(R)$ .

Lemma 3 Let $T \in B (H)$ and $F$ is a power finite rank operator with $F T = T F$ . The following statements are equivalent:

(1) $T + F$ satisfies property $(W_{E})$ and is an isoloid operator;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T)$ ;

(3) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{e a} (T) ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ .

Proof (1) $\Rightarrow$ (2). Since $T + F$ satisfies property $(W_{E})$ and $σ_{w} (T + F) = σ_{w} (T)$ , $σ_{b} (T + F) =$ $σ_{b} (T)$ , we have that $σ_{w} (T) = σ_{b} (T)$ . If $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T)$ , when $λ_{0} \notin σ_{τ} (T)$ , by the proof of Lemma 1, we know that $T - λ_{0} I$ is Drazin invertible. Thus $λ_{0} \in i s o σ (T + F) ⋃ ρ (T + F)$ . If $n (T + F - λ_{0} I) = \infty$ , then $λ_{0} \in E (T + F)$ . Since $T + F$ has property $(W_{E})$ , we get that $T + F - λ_{0} I$ is a Browder operator. It is a contradiction. Therefore $n (T + F - λ_{0} I) < \infty$ , hence $n (T - λ_{0} I) < \infty$ . By $T - λ_{0} I$ is Drazin invertible we have that $λ_{0} \notin σ_{b} (T)$ . Suppose that $λ_{0} \notin a c c σ (T)$ . Without loss of generality, we assume that $λ_{0} \in i s o σ (T)$ . Then $λ_{0} \in i s o σ (T + F)$ . The fact that $T + F$ is isoloid implies that $λ_{0} \in E (T + F)$ . Because $T + F$ has the property $(W_{E})$ , $T + F - λ_{0} I$ is a Browder operator. We also have that $λ_{0} \notin σ_{b} (T)$ .

(2) $\Rightarrow$ (1). Suppose that $λ_{0} \in σ (T + F) \ σ_{w} (T + F)$ . Then we have that $T - λ_{0} I$ is a Weyl operator. So $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T)$ . By condition (2), we can get that $T - λ_{0} I$ is a Browder operator. Thus $T + F - λ_{0} I$ is a Browder operator. Then $λ_{0} \in E (T + F)$ . Contrarily, if $λ_{0} \in E (T + F)$ , then $λ_{0} \in ρ (T)$ $⋃ i s o σ (T)$ . Therefore $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T)$ . Hence $T - λ_{0} I$ and $T + F - λ_{0} I$ are Browder operators.

If there exists $λ_{0} \in i s o σ (T + F)$ such that $n (T + F - λ_{0} I) = 0$ , then $λ_{0} \in i s o σ (T) ⋃ ρ (T)$ . By condition (2), we have that $T - λ_{0} I$ is a Browder operator. It follows that $T + F - λ_{0} I$ is a Browder operator. The fact that $n (T + F - λ_{0} I) = 0$ implies that $T + F - λ_{0} I$ is invertible, which is a contradiction. Therefore $T + F$ is an isoloid operator.

(1) $\Rightarrow$ (3). Suppose $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{e a} (T) ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ . If $λ_{0} \notin σ_{τ} (T)$ , from the proof of (1) $\Rightarrow$ (2) we can get that $λ_{0} \notin σ_{b} (T)$ . If $λ_{0} \notin a c c σ_{a} (T)$ , we assume that $λ_{0} \in σ (T)$ without loss of generality. Since $λ_{0} \notin a c c σ_{e a} (T) ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ , we have that $λ_{0} \in i s o σ (T)$ . Hence $λ_{0} \in i s o σ (T + F)$ . Since $T + F$ is isoloid and satisfies property $(W_{E})$ , we have that $T + F - λ_{0} I$ is a Browder operator. Thus $λ_{0} \notin σ_{b} (T)$ .

(3) $\Rightarrow$ (1). The proof for (3) $\Rightarrow$ (1) is the same as the proof for (2) $\Rightarrow$ (1).

Lemma 4 Let $T \in B (H)$ and $F$ is a power finite rank operator with $F T = T F$ . The following statements are equivalent:

(1) $T + F$ is an isoloid operator and satisfies property $(R)$ , $σ_{a} (T + F) = σ (T + F)$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ ;

(3) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)] .$

Proof (1) $\Rightarrow$ (2). Suppose $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty} ⋃$ $[a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . We assume that $λ_{0} \in σ (T)$ . If $λ_{0} \in ρ_{τ} (T)$ , when $λ_{0} \notin a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ , we can get that $λ_{0} \in \partial σ (T)$ . Then $T - λ_{0} I$ is a Browder operator. When $λ_{0} \notin σ_{c} (T)$ , we have that $T - λ_{0} I$ is a Weyl operator. Hence $λ_{0} \in ρ_{w} (T + F)$ . The fact that $T + F$ satisfies property $(R)$ implies that $T + F - λ_{0} I$ is a Browder operator. Hence $λ_{0} \in ρ_{b} (T)$ . If $λ_{0} \notin$ $a c c σ (T)$ , then $λ_{0} \in i s o σ (T + F) ⋃ ρ (T + F)$ . We can assume that $λ_{0} \in i s o σ (T + F)$ . Since $T + F$ is an isoloid operator, we have that $λ_{0} \in π_{00} (T + F)$ . Since $T + F$ satisfies property $(R)$ , we can get that $T + F - λ_{0} I$ is a Browder operator. Therefore $λ_{0} \in ρ_{b} (T)$ .

(2) $\Rightarrow$ (1). Let's first prove that $T + F$ is an isoloid operator. If there exists $λ_{0} \in i s o σ (T + F)$ such that $n (T + F - λ_{0} I) = 0$ . We assume that $λ_{0} \in σ (T)$ without loss of generality. Thus $n (T - λ_{0} I) < \infty$ , $λ_{0} \in i s o σ (T)$ . Therefore $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty} ⋃$ $[a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . It follows that $T - λ_{0} I$ is a Browder operator. Then $T + F - λ_{0} I$ is a Browder operator. Hence $T + F - λ_{0} I$ is invertible, which is a contradiction. Thus $T + F$ is an isoloid operator.

Next we will prove $σ_{a} (T + F) = σ (T + F)$ . If $λ_{0} \in ρ_{a} (T + F)$ , then $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)]$ $⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . It implies that both $T - λ_{0} I$ and $T + F - λ_{0} I$ are Browder operators. Hence $λ_{0} \notin σ (T + F)$ .

If $λ_{0} \in σ_{a} (T + F) \ σ_{a b} (T + F)$ , then $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T -$ $λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Hence $T - λ_{0} I$ and $T + F - λ_{0} I$ are Browder operators. Thus $λ_{0} \in π_{00} (T + F)$ . Conversely, suppose that $λ_{0} \in π_{00} (T + F)$ , then $λ_{0} \in i s o σ (T) ⋃ ρ (T)$ . Then we also have that $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty}$

$⋃ [a c c {λ \in C :$ $n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Thus $T - λ_{0} I$ and $T + F - λ_{0} I$ are Browder operators. Hence $T + F$ satisfies property $(R)$ .

The proof for (1) $\Leftrightarrow$ (3) is the same as the proof for (1) $\Leftrightarrow$ (2).

Then we obtain the finite rank perturbation with two properties held at the same time.

Theorem 2 Let $T \in B (H)$ and $F$ is a power finite rank operator with $F T = T F$ . The following statements are equivalent:

(1) $T + F$ is an isoloid operator and satisfies property $(R)$ and property $(W_{E})$ , $σ_{a} (T + F)$ $= σ (T + F)$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in a c c σ (T) : n (T - λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂$

$σ_{c} (T)]$

Proof (1) $\Rightarrow$ (2). Suppose $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in a c c σ (T) : n (T$ $- λ I) = \infty} ⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . If $λ_{0} \notin a c c σ_{a b} (T)$ , by Lemma 4 we have that $λ_{0} \notin σ_{b} (T)$ . If $λ_{0} \notin σ_{w} (T)$ , by Lemma 3 we also get that $λ_{0} \notin σ_{b} (T)$ .

(2) $\Rightarrow$ (1). The statement (2) implies that $σ_{w} (T) = σ_{b} (T)$ . Suppose $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)]$ $⋃ a c c σ_{w} (T)$ , then $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ [a c c {λ \in C : n (T - λ I) <$ $d (T - λ I)} ⋂ σ_{c} (T)]$ . If $λ_{0} \notin a c c σ (T)$ , then $λ_{0} \notin {λ \in a c c$

$σ (T) : n (T - λ I) = \infty}$ . If $λ_{0} \in ρ_{τ} (T)$ , we have that $λ_{0} \in ρ (T) ⋃ \partial σ (T)$ . Thus $T - λ_{0} I$ is Drazin invertible. Therefore $λ_{0} \notin {λ \in$ $a c c σ (T) : n (T - λ I) = \infty}$ . Hence $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)] ⋃ {λ \in$ $a c c σ (T) :$ $n (T - λ I) = \infty} ⋃ [a c c$ ${λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Then $T - λ_{0} I$ is a Browder operator. Thus $σ_{b} (T) = [σ_{τ} (T) ⋂$ $a c c σ (T)] ⋃ a c c σ_{w} (T)$ . With Lemma 3 we get that $T + F$ is isoloid and satisfies property $(W_{E})$ .

By condition (2) we have that $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C :$ $n (T - λ I) = \infty}$ $⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ . Then according to Lemma 4 we have that $T + F$ satisfies property $(R)$ and $σ_{a} (T + F) = σ (T + F)$ . Therefore (1) holds.

3 Property $(W_{E})$ and Property $(R)$ for Operator Functions

Let $H (T)$ be the class of all complex-valued functions which are analytic on a neighborhood of $σ (T)$ and are not constant on any component of $σ (T)$ . For the function's properties, we discuss property $(W_{E})$ first.

Lemma 5 Let $T \in B (H)$ . For any $f \in H (T)$ , $f (T)$ satisfies property $(W_{E})$ if and only if $\forall λ$ , $μ \in ρ_{e} (T)$ , $i n d (T - λ I) \cdot i n d (T - μ I) \geq 0$ and one of the following conditions holds:

(1) $σ (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0}$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T)$ .

Proof " $\Leftarrow$ ". Suppose (1) holds. Then $σ_{0} (T) = \emptyset$ , $σ (T) = σ_{b} (T) = σ_{w} (T)$ , thus for any $f \in H (T)$ , $σ (f (T)) = f (σ (T)) = f (σ_{w} (T)) = σ_{w} (f (T))$ , $σ (f (T)) \ σ_{w} (f (T)) = \emptyset$ . Since $E (f (T)) \subseteq$ $f (E (T))$ , we have that $E (f (T)) = \emptyset$ . So $f (T)$ satisfies property $(W_{E})$ . If (2) holds, then $i s o σ (T)$ $\subseteq σ_{0} (T)$ , $σ_{w} (T) = σ_{b} (T)$ . Hence $σ_{w} (f (T)) = f (σ_{w} (T)) = f (σ_{b} (T)) = σ_{b} (f (T))$ . It follows that $σ (f (T)) \ σ_{w} (f (T)) \subseteq σ_{0} (f (T)) \subseteq E (f (T))$ . For the converse, let $μ_{0} \in E (f (T))$ , and let $f (T) -$ $μ_{0} I = (T - λ_{1} {I)}^{n_{1}} (T - λ_{2} {I)}^{n_{2}} \dots (T - λ_{t} {I)}^{n_{t}} g (T)$ , where $λ_{i} \neq λ_{j} (i, j = 1,2, \dots, t)$ , and $g (T)$ is inverse. We assume that $λ_{i} \in σ (T)$ , then $λ_{i} \in i s o σ (T)$ , thus $λ_{i} \in σ_{0} (T)$ . It follows that $f (T) - μ_{0} I$ is a Browder operator.

" $\Rightarrow$ ". If there exists $λ_{1}, λ_{2} \in ρ_{e} (T)$ such that $i n d (T - λ_{1} I) = n > 0$ , $i n d (T - λ_{2} I) = - m < 0$ , suppose that $f_{1} (T) = (T - λ_{1} {I)}^{m} (T - λ_{2} {I)}^{n}$ , then $0 \in σ (f_{1} (T)) \ σ_{w} (f_{1} (T))$ . By $f_{1} (T)$ satisfies property $(W_{E})$ , we have that $f_{1} (T)$ is a Browder operator. Hence both $T - λ_{1} I$ and $T - λ_{2} I$ are Browder operators. It is a contradiction. Thus $\forall λ, μ \in ρ_{e} (T)$ , $i n d (T - λ I) \cdot i n d (T - μ I) \geq 0$ .

If $σ_{0} (T) = \emptyset$ , then the fact that $T$ satisfies property $(W_{E})$ implies that $σ (T) = σ_{w} (T) = σ_{b} (T)$ , $E (T) = \emptyset$ . According to Lemma 1, we can get (1) holds. When $σ_{0} (T) \neq \emptyset$ , we can assert that ${λ \in$ $i s o σ (T) : n (T - λ I) = 0} = \emptyset$ . If not, suppose that $λ_{3} \in {λ \in i s o σ (T) : n (T - λ I) = 0}$ , $λ_{4} \in σ_{0} (T)$ . Let $f_{2} (T) = (T - λ_{3} I) (T - λ_{4} I)$ , then $0 \in E (f_{2} (T))$ . Since $f_{2} (T)$ satisfies property $(W_{E})$ , then $f_{2} (T)$ is a Browder operator, so is $T - λ_{3} I$ , which is a contradiction. Therefore ${λ \in i s o σ (T) : n (T - λ I) = 0} = \emptyset$ . By Lemma 1 and its proof, we get that (2) holds.

Similarly, we can obtain the following conclusion. Let $T \in B (H)$ , for any $f \in H (T)$ , $f (T)$ satisfies property $(W_{E})$ if and only if $\forall λ, μ \in ρ_{e} (T)$ , $i n d (T - λ I) \cdot i n d (T - μ I) \geq 0$ and one of the following conditions holds:

(1) $σ (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{e a} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ a c c {λ \in C : n (T - λ I) <$ $d (T - λ I)}$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ_{a} (T)] ⋃ a c c σ_{e a} (T) ⋃ a c c {λ \in C : n (T - λ I) < d (T - λ I)}$ .

Corollary 3 Let $T \in B (H)$ , for any $f \in H (T)$ , $f (T)$ satisfies property $(W_{E})$ if and only if $\forall λ, μ \in ρ_{e} (T)$ , $i n d (T - λ I) \cdot i n d (T - μ I) \geq 0$ and the following conditions holds:

(1) $T$ satisfies property $(W_{E})$ ;

(2) If $σ_{0} (T) \neq \emptyset$ , then $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{w} (T)$ .

For property $(R)$ of operator functions, we have the following conclusions.

Lemma 6 Let $T \in B (H)$ , for any $f \in H (T)$ , $f (T)$ satisfies property $(R)$ if and only if one of the following conditions holds:

(1) $σ (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T - λ I) = \infty}$ $⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty}$ .

Proof " $\Rightarrow$ ". If $σ_{0} (T) = \emptyset$ , then $σ (T) = σ_{b} (T) = σ_{w} (T)$ . By Lemma 2, we can get that (1) holds.

If $σ_{0} (T) \neq \emptyset$ , we can assert that $σ (T) = σ_{a} (T)$ . If not, suppose that $λ_{1} \in ρ_{a} (T) ⋂ σ (T)$ , $λ_{2} \in σ_{0} (T)$ , let $f (T) = (T - λ_{1} I) (T - λ_{2} I)$ , then $0 \in σ_{a} (f (T)) \ σ_{a b} (f (T))$ . Since $f (T)$ satisfies property $(R)$ , we can get that $f (T)$ is a Browder operator. Then $T - λ_{1} I$ is a Browder operator, which is a contradiction. Therefore $σ (T) = σ_{a} (T)$ . Similarly, we have ${λ \in i s o σ (T) : n (T - λ I) = 0} = \emptyset$ in this situation. Suppose that $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty}$ . If $λ_{0} \notin σ_{τ} (T)$ , by $T$ satisfies property $(R)$ and $σ (T) = σ_{a} (T)$ we have that $λ_{0} \in \partial σ (T)$ . The fact that $n (T - λ_{0} I) < \infty$ implies that $T - λ_{0} I$ is a Browder operator. If $λ_{0} \notin a c c σ (T)$ , we can assume that $λ_{0} \in σ (T)$ . Since ${λ \in i s o σ (T) :$ $n (T - λ I) = 0} = \emptyset$ , we have $λ_{0} \in π_{00} (T)$ . By $T$ has property $(R)$ , we can get that $T - λ_{0} I$ is a Browder operator. Therefore (2) holds.

" $\Leftarrow$ ". If condition (1) holds, then $σ_{0} (T) = \emptyset$ , $π_{00} (T) = \emptyset$ , $σ_{a} (T) = σ_{a b} (T)$ . Then for any $f \in$ $H (T)$ , $σ_{a} (f (T)) = f (σ_{a} (T)) = f (σ_{a b} (T)) = σ_{a b} (f (T))$ . Hence $σ_{a} (f (T)) \ σ_{a b} (f (T)) = \emptyset$ . Since $π_{00} (f (T)) \subseteq f (π_{00} (T))$ , we have that $π_{00} (f (T)) = \emptyset$ . Therefore $f (T)$ satisfies property $(R)$ .

If conditions (2) holds, then $σ_{a b} (T) = σ_{b} (T)$ , $σ_{a} (T) = σ (T)$ , ${λ \in i s o σ (T) :$ $n (T - λ I) = 0} = \emptyset$ . Hence $σ_{a} (f (T)) \ σ_{a b} (f (T)) \subseteq σ_{0} (f (T)) \subseteq π_{00} (f (T))$ . Suppose that $μ_{0} \in π_{00} (f (T))$ . Let $f (T) -$ $μ_{0} I = (T - λ_{1} {I)}^{n_{1}} (T - λ_{2} {I)}^{n_{2}} \dots (T - λ_{t} {I)}^{n_{t}} g (T)$ , where $λ_{i} \neq λ_{j}$ , $g (T)$ is invertible. We can assume that $λ_{i} \in σ (T)$ , then $λ_{i} \in π_{00} (T)$ . Since $T$ has property $(R)$ , we get $λ_{i} \in ρ_{b} (T)$ . So $f (T) - μ_{0} I$ is a Browder operator. Therefore $f (T)$ satisfies property $(R)$ .

By the proof of Lemma 6, we can get that:

Corollary 4 Let $T \in B (H)$ . For any $f \in H (T)$ , $f (T)$ satisfies property $(R)$ if and only if the following conditions holds:

(1) $T$ satisfies property $(R)$ ;

(2) If $σ_{0} (T) \neq \emptyset$ , then $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty}$ .

Then we obtain the condition that the operator function satisfies both two properties.

Theorem 3 Let $T \in B (H)$ . For any $f \in H (T)$ , $f (T)$ satisfies property $(R)$ and property $(W_{E})$ if and only if $\forall λ, μ \in ρ_{e} (T)$ , $i n d (T - λ I) \cdot i n d (T - μ I) \geq 0$ and:

(1) $T$ satisfies property $(R)$ and property $(W_{E})$ ;

(2) If $σ_{0} (T) \neq \emptyset$ , then $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)]$ .

Proof " $\Rightarrow$ ". According to Corollary 3 and Corollary 4, we know that we only need to prove (2). Suppose that $λ_{0} \notin [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)]$ . If $λ_{0} \notin σ_{w} (T)$ , by Corollary 3 we know that $T - λ_{0} I$ is a Browder operator. Suppose $λ_{0} \notin a c c σ_{a b} (T)$ . When $n (T - λ_{0} I) = 0$ , by Corollary 4 we can get that $λ_{0} \in ρ_{b} (T)$ . When $n (T - λ_{0} I) > 0$ , the fact that $σ (T) = σ_{a} (T)$ implies $λ_{0} \in i s o σ (T)$ . Thus $λ_{0} \in E (T)$ . We also get that $T - λ_{0} I$ is a Browder operator. The converse is obviously true. Hence (2) holds.

The sufficiency is easy to get from Corollary 3 and Corollary 4.

Corollary 5 Let $T \in B (H)$ . For any $f \in H (T)$ , $f (T)$ satisfies property $(R)$ and property $(W_{E})$ if and only if $\forall λ, μ \in ρ_{e} (T)$ , $i n d (T - λ I) \cdot i n d (T - μ I) \geq 0$ and one of the following conditions holds:

(1) $σ (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in σ (T) : n (T - λ I) = 0} ⋃ {λ \in C : n (T -$ $λ I) = \infty}$ $⋃ [a c c {λ \in C : n (T - λ I) < d (T - λ I)} ⋂ σ_{c} (T)]$ ;

(2) $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ a c c σ_{a b} (T) ⋃ {λ \in C : n (T - λ I) = \infty}$ .

Proof " $\Rightarrow$ ". If $σ_{0} (T) = \emptyset$ , then $σ (T) = σ_{w} (T) = σ_{b} (T)$ . Since ${λ \in C : n (T - λ I) = \infty}$ $= {λ \in a c c σ (T) : n (T - λ I) = \infty}$ , by Theorem 1 and Remark 2 we can get that (1) holds. If $σ_{0} (T) \neq \emptyset$ , the condition (2) hold.

" $\Leftarrow$ ". According to Theorem 1 and Theorem 3, we know that for any $f \in H (T)$ , $f (T)$ satisfies property $(R)$ and property $(W_{E})$ .

Example 1 Let $A, B \in B (l^{2})$ be defined by: $A (x_{1}, x_{2}, \dots) = (x_{2}, x_{3}, \dots)$ , $B (x_{1}, x_{2}, x_{3}, \dots)$ $= (x_{1}, 0,0, 0, \dots)$ . Suppose that $T = (\begin{matrix} A & 0 \\ 0 & I + B \end{matrix})$ . Then: (1) $T$ satisfies property $(R)$ and property $(W_{E})$ ; (2) For any $λ \in ρ_{e} (T)$ , $i n d (T - λ I) \geq 0$ ; (3) $σ_{0} (T) \neq \emptyset$ , and $σ_{b} (T) = [σ_{τ} (T) ⋂ a c c σ (T)] ⋃ [a c c σ_{a b} (T) ⋂ σ_{w} (T)]$ .

Therefore by Theorem 3, we can get that for any $f \in H (T)$ , $f (T)$ satisfies property $(R)$ and property $(W_{E})$ .

References

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[1] Djordjević D S. Operators obeying a-Weyl's theorem[J]. Publicationes Mathematicae Debrecen, 1999, 55(3/4): 283-298. [Google Scholar]

[2] Cao X H, Guo M Z, Meng B. Weyl spectra and Weyl's theorem[J]. Journal of Mathematical Analysis and Applications, 2003, 288(2): 758-767. [Google Scholar]

[3] Gupta A, Kumar A. Properties (BR) and (BgR) for bounded linear operators[J]. Rendiconti Del Circolo Matematico Di Palermo Series 2, 2020, 69(2): 601-611. [Google Scholar]

[4] Sun C H, Wang N, Cao X H. Topological uniform descent and judgement of a-Weyl's theorem [J]. Wuhan University Journal of Natural Sciences, 2023, 28(5): 392-398. [Google Scholar]

[5] Aiena P, Triolo S. Weyl-type theorems on Banach spaces under compact perturbations[J]. Mediterranean Journal of Mathematics, 2018, 15(3): 126. [Google Scholar]

[6] Wu X F, Huang J J, Chen A. Weylness of 2 × 2 operator matrices[J]. Mathematische Nachrichten, 2018, 291(1): 187-203. [Google Scholar]

[7] Dong J, Cao X H, Dai L. On weyl's theorem for functions of operators[J]. Acta Mathematica Sinica, English Series, 2019, 35(8): 1367-1376. [CrossRef] [MathSciNet] [Google Scholar]

[8] Dai L, Yi J L. Property (ω) and Hypercyclic Property for Operators[J]. Wuhan University Journal of Natural Sciences, 2024, 29(6): 499-507. [Google Scholar]

[9] Zhu S, Li C G, Zhou T T. Weyl type theorems for functions of operators [J]. Glasgow Mathematical Journal, 2012, 54(3): 493-505. [Google Scholar]

[10] Berkani M, Kachad M. New Browder and weyl type theorems[J]. Bulletin of the Korean Mathematical Society, 2015, 52(2): 439-452. [CrossRef] [MathSciNet] [Google Scholar]

[11] Yang L L, Cao X H. Property (R) for functions of operators and its perturbations[J]. Mediterranean Journal of Mathematics, 2022, 19(1): 25. [Google Scholar]

[12] Ren Y X, Jiang L N, Kong Y Y. Property and topological uniform descent [J]. Bulletin of the Belgian Mathematical Society - Simon Stevin, 2022, 29(1): 1-17. [MathSciNet] [Google Scholar]

[13] Grabiner S. Uniform ascent and descent of bounded operators[J]. Journal of the Mathematical Society of Japan, 1982, 34(2): 317-337. [Google Scholar]

[14] Rakočević V. Semi-Browder operators and perturbations[J]. Studia Mathematica, 1997, 122(2): 131-137. [Google Scholar]

Property ( W E ) and Property ( R ) for Operator and Its Functions

算子及算子函数的 ( W E ) 性质与 ( R ) 性质