Issue 
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 5, October 2023



Page(s)  392  398  
DOI  https://doi.org/10.1051/wujns/2023285392  
Published online  10 November 2023 
Mathematics
CLC number: O177
Topological Uniform Descent and Judgement of AWeyl's Theorem
^{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
In this paper, aBrowder's theorem and aWeyl's theorem for bounded linear operators are studied by means of the property of the topological uniform descent. The sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space holding aBrowder's theorem and aWeyl's theorem are established. As a consequence of the main result, the new judgements of aBrowder's theorem and aWeyl's theorem for operator function are discussed.
Key words: aBrowder's theorem / aWeyl's theorem / topological uniform descent
Biography: SUN Chenhui, female, Ph. D., Assistant professor, research direction: operator theory. Email: sunchenhui1986@163. com
Fundation item: Supported by the 2021 General Special Scientific Research Project of Education Department of Shaanxi Provincial Government (21JK0637), Science and Technology Planning Project of Weinan Science and Technology Bureau (2022ZDYFJH11), and 2021 Talent Project of Weinan Normal University (2021RC16)
© Wuhan University 2023
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 research of Weyl type theorem is an important subject in spectral theory. In 1909, Weyl discovered Weyl's theorem when he studied the spectrum of the selfadjoint operator^{[1]}. Then Harte and Lee defined Browder's theorem^{[2]}. Rakočević gave two other variations of Weyl's theorem: aWeyl's theorem and aBrowder's theorem^{[3,4]}. These generalizations are called Weyl type theorems by scholars. The study on Weyl type theorems can well reflect the structural characteristics of spectrums. Hence the research of Weyl type theorem has attracted much attention and got many good results in resent years^{[57]}. In this paper, we mainly study aBrowder's theorem and aWeyl's theorem for bounded linear operators and operator functions by means of the property of the topological uniform descent.
In this paper, denotes a complex separable infinite dimensional Hilbert space. Let be the algebra of all bounded linear operators on . For an operator we shall denote by the dimension of the kernel , and by the codimension of the range . We call is an upper semiFredholm operator if is closed and . We say that is a lower semiFredholm operator when . An operator is said to be Fredholm if is closed and both and are finite. If is an upper (or a lower) semiFredholm operator, the index of , , is defined to be . The ascent of , , is the least nonnegative integer such that and the descent, , is the least nonnegative integer such that . The operator is Weyl if it is Fredholm of index zero, and is said to be Browder if it is Fredholm "of finite ascent and descent". We call a Drazin invertible operator if . Let be the spectrum of . The approximate point spectrum of is denoted by . The Weyl spectrum , the upper semiFredholm spectrum , the Browder spectrum , the Drazin spectrum , are defined by is not Weyl( denotes the set of complex numbers ), is not upper semiFredholm, is not Browder, is not Drazin invertible. Let , , , , , . For a set , we write () for the set of isolated (accumulation) points of , and we denote () for interior (boundary) points set of . If , then is a pole of . is called an aisoloid operator if , where .
satisfies aBrowder's theorem if
where is not upper semiFredholm or and is not upper semiFredholm or . Let and . The aWeyl's theorem holds for if and only if
where we write . It can be shown that aWeyl's theorem aBrowder's theorem, but the converse is not true. Let be defined by . Then and . So satisfies aBrowder's theorem, but aWeyl's theorem does not hold for .
If satisfies , then is called a Sapher operator^{[8,9]}. The Sapher spectrum is is not Sapher operator. Goldberg defined is not closed^{[10]}. is called a Kato operator if is closed and . Therefore, the Kato spectrum is .
Let , for each nonnegative integer , induces a linear transformation from the vector space to . We denote by the dimension of the null space of the induced map and put . If there is a nonnegative integer for which for and is closed in the operator range topology of for , then we say that has topological uniform descent^{[11]}. If is upper semiFredholm, then has topological uniform descent. Let has topological uniform descent, and . We will use the following property which is discovered by Grabiner (Ref.[11], Corollary 4.9):
Lemma 1 Let , . If has topological uniform descent, then .
On the basis of analyzing distribution of various spectrums of bounded linear operators, the sufficient and necessary conditions holding aBrowder's theorem and aWeyl's theorem are established by means of the property of the topological uniform descent. In addition, the new judgements of aBrowder's theorem and aWeyl's theorem for operator function are discussed.
1 Judgement of ABrowder's Theorem and AWeyl's Theorem for Bounded Linear Operator
First, we describe aBrowder's theorem by the relation between topological uniform descent and .
Theorem 1 satisfies aBrowder's theorem if and only if .
Proof "". Suppose
Then there exists a deleted neighborhood centered on such that for any , . Moreover, for any deleted neighborhood , there exists such that . Let , then we will get that is Browder operator since satisfies aBrowder's theorem and . It follows that . Since , , we know that according to Lemma 1.
"". It's clear that
Suppose . According to perturbation theorem of semiFredholm operator, there exists such that if . Then . It follows that . If and , then is Browder operator. Therefore, aBrowder's theorem holds for .
Remark 1 (i) In Theorem 1, suppose satisfies aBrowder's theorem, then each part of the decomposition of cannot be deleted.
(a) Let be defined by . Then , satisfies aBrowder's theorem.
But . Thus cannot be deleted.
(b) Let be defined by . We can get that , aBrowder's theorem holds for . However, ,
which means cannot be deleted.
(c) Let be defined by . Then . satisfies aBrowder's theorem. But . Therefore cannot be deleted.
(d) Let be defined by . We have , which implies aBrowder's theorem holds for . Since , cannot be deleted.
(ii) Since , satisfies aBrowder's theorem if and only if .
(iii) By Theorem 1, if , then satisfies aBrowder's theorem. But the converse is not true. Let be defined by . We can get that aBrowder's theorem holds for , but .
(iv) satisfies aBrowder's theorem and .
In fact, yields .
For the converse, since , by Theorem 1 we get .
(v) satisfies aBrowder's theorem and , where is denumerable.
"". It is clear that . Then we can get that satisfies aBrowder's theorem by Theorem 1. Since , we have that . Since , it follows that is denumerable.
"". Suppose that . If , then by aBrowder's theorem holds for . If , then there exists a neighborhood centered on such that . Since is denumerable, for any , there exists such that is Weyl operator. It follows that since satisfies aBrowder's theorem. We can also get .
By Theorem 1, the following results can be obtained.
Corollary 1 Let . The following statements are equivalent:
(1) satisfies aBrowder's theorem;
(2) ;
(3) ;
(4) ;
(5) .
Proof (1)(2). Using Theorem 1, we have when satisfies aBrowder's theorem. Since , it follows that (2) holds.
(2)(3). Let and is Kato operator. Then (Ref. [12], Lemma 3.4). Therefore . By (2) we know that (3) holds.
(3)(4). Suppose . Then there exists a deleted neighborhood centered on such that for any , . Since , for any , there exists such that . Let , it follows that
Thus we can get is Browder operator by (3). From the proof of Theorem 1, we have .
(4)(5). Let . Then there exists a deleted neighborhood centered on such that for any , and . Since , we know that for any , there exists
It follows that is Browder operator. This implies that (5) holds.
(5)(1). It is clear that
If , then , therefore . If and , then is Browder operator. We can conclude that satisfies aBrowder's theorem.
Corollary 2 Let . The following statements are equivalent:
(1) satisfies aBrowder's theorem;
(2) ;
(3)
Proof (1)(2). We only need to prove
.Suppose . Then there exists a deleted neighborhood centered on such that for any , . If , there exists such that for any , . Let . It follows that for any , is Weyl operator. From the aBrowder's theorem holds for and the proof of Theorem 1, we can get that is Browder operator. If , then there exists a deleted neighborhood centered on such that for any , since satisfies aBrowder's theorem. Therefore . Moreover, by and , we know that .
(2)(3). It is clear that . Since , and , we can get that . Also, . Then we have
Hence (2)(3) is true.
(3)(1). We know that , and . If , then . If and , then is Browder operator. Thus satisfies aBrowder's theorem.
In the following, we will discuss the aWeyl's theorem for .
Theorem 2 Let . The following statements are equivalent:
(1) satisfies aWeyl's theorem;
(2)
;
(3) .
Proof (1)(2). Since satisfies aWeyl's theorem, we know that , . From Corollary 2, we have . Moreover, . Since . We get .
Also, . Hence . Then we know that (2) holds.
(2)(3). Since and , it follows that .
(3)(1). , , . If and , then is Browder operator. Moreover, . Therefore . Similarly, we can prove that . So the aWeyl's theorem holds for .
Remark 2 (i) By Theorem 2, we can get that if , then satisfies aWeyl's theorem. But the converse is not true. Let be defined by . Then satisfies aWeyl's theorem, but .
(ii) satisfies aWeyl's theorem, and .
In fact, suppose that . Then , so . Similarly, since and , we know that .
For the converse, let . If , then is invertible. If , we can get that is Browder operator by and the aWeyl's theorem holds for . If , by we can get is Browder operator.
2 Judgement of ABrowder's Theorem and AWeyl's Theorem for Operator Function
In the following, we will research the aBrowder's theorem and aWeyl's theorem for operator function by means of the property of the topological uniform descent.
Theorem 3 Let . Then for any polynomial , satisfies aBrowder's theorem if and only if
(1) satisfies aBrowder's theorem;
(2) If , then .
Proof Suppose satisfies aBrowder's theorem for any polynomial . We only need to prove (2) holds. We can assert that if , then for any , . In fact, suppose there exists , where is finite or . Let , . We can see that is finite. If , let . Moreover, let if . Then . It follows that . It is in contradiction to the fact that . Then we will prove that . If and , then for any deleted neighborhood centered on , there exists such that . Since for any , , we know that is Weyl operator. By satisfing aBrowder's theorem, we can get is Browder operator. Thus . From and , we conclude that is Browder operator.
For the converse, if , then for any , . If , we can get that since but . Let and , where and , . It follows that is upper semiFredholm operator and for all . From satisfing aBrowder's theorem, we have that . Hence , satisfies aBrowder's theorem.
By Theorem 3 and the proof procedure, we can get the following results.
Corollary 3 Let . Then for any polynomial , satisfies aBrowder's theorem if and only if
(1) satisfies aBrowder's theorem;
(2) If , then .
Remark 3 (i) Suppose , then satisfies aBrowder's theorem and . This implies that for any polynomial , satisfies aBrowder's theorem. However, the converse is not true. Let . The aBrowder's theorem holds for , but .
for any polynomial , satisfies aBrowder's theorem, .
In fact, suppose . This implies that . So .
For the converse, let and . Then for any deleted neighborhood centered on , there exists such that . Thus is Browder operator. So . By and , we know that is Browder operator.
(ii) for any polynomial , satisfies aBrowder's theorem, .
In the following, we will establish sufficient and necessary conditions for operator functions holding aWeyl's theorem.
Theorem 4 Let . Then is aisoloid and for any polynomial , satisfies aWeyl's theorem if and only if:
(1)
;
(2) If , then .
Proof "". First, we will prove that is aisoloid. If there exists and , then . If , then . It is a contradiction. If , then is Browder operator, so . It is a contradiction too. Thus is aisoloid. From (1) we know that satisfies aBrowder's theorem. By Theorem 3, we have that for any polynomial , satisfies aBrowder's theorem. Suppose , let , where and , . Because is aisoloid, we get that . From (1) and the proof of Theorem 3, we have . Then , hence . Thus for any polynomial , satisfies aWeyl's theorem.
"". By Theorem 3 we know (2) holds. Suppose . If , then . In fact, if , from is aisoloid we can get that . Hence . Since , it follows that is Browder operator. If , from the proof of Theorem 2, we get (1) holds.
Remark 4 (i) By Theorem 2, we can get that if , then for any polynomial , satisfies aWeyl's theorem. But from Remark 2(ii), we know that the converse is not true. However,
for any polynomial , satisfies aWeyl's theorem, and .
(ii) Let . If , then for any polynomial , satisfies aWeyl's theorem. In fact, if , let , where and , . Then . Since , we have . By we know that is Browder operator. Thus is Browder operator. By Remark 3, we can get that satisfies aWeyl's theorem for any polynomial . But the converse is not true. Let . Then for any polynomial , satisfies aWeyl's theorem. However, .
if and only if for any polynomial , satisfies aWeyl's theorem, is aisoloid and .
In fact, if , we only need to prove is aisoloid. If and . Then . It follows that is Browder operator. So is invertible. It is in contradiction to the fact that .
For the converse, we only need to prove that . Since is aisoloid, this implies that . By aWeyl's theorem holds for , we get that .
Corollary 4 Let . Then is aisoloid and for any polynomial , satisfies aWeyl's theorem if and only if:
(1) ;
(2) , .
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