New Decompositions for Classes of Operators with Topological Uniform Descent

第 55 卷 第 6 期

2020 年 12 月

JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY

Vol. 55 No. 6

Dec. 2020

ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.6.31

Mathematics

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混合資源和相互作用模式對偏遠地區可再生能源管理的影響

Orlando García a, _{*, Carlos Carpintero} a_{, José Sanabria} b_{, Osmin Ferrer} b

a _{Corporación Universitaria del Caribe-CECAR, Departamento de Ciencias Básicas, Sincelejo, Colombia.}

ogarciam554@gmail.com, carpintero.carlos@gmail.com

b

Universidad de Sucre, Departamento de Matemáticas, Sincelejo, Colombia, jesanabri@gmail.com,

osmin.ferrer@unisucre.edu.co

Received: June 13, 2020 ▪ Review: September 8, 2020 ▪ Accepted: October 9, 2020

This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

Abstract

The article describes a new decomposition property for operators with topological uniform descent, like Kato type operators, as well as new results on the stability of this class of operators under perturbations by operators with finite-range power based on topological descent notion, from which we can generalize many perturbation results for a large classes of operators by extending to Banach spaces known techniques on Hilbert spaces. As application of our resuts we obtain that 𝑇 is a lower semi B-Weyl

operator if and only if 𝑇 = 𝑆 + 𝐾 , where 𝑆 is a lower semi B-Browder operator and dim 𝐾𝑛_{(𝑋) < ∞, for}

some 𝑛 ∈ ℕ. Our methods generalize to Banach spaces some results obtained by Aiena for operators acting on Hilbert spaces.

Keywords: Quasi-Fredholm Operator, Operators with Topological Uniform Descent, Operators with

Finite-range Power. 摘要 本文介紹了具有拓撲均勻下降的算子（如加藤型算子）的新分解性質，以及此類算子在拓撲 下降概念的作用下，具有有限範圍冪的算子在擾動下的穩定性的新結果。通過將希爾伯特空間上 的已知技術擴展到巴纳赫空間，可以將一類大型算子的許多攝動結果推廣化。作為我們的結果的 應用，當且僅當 T =小号+ K 時，T 是下半魏尔算符，其中小号是下半乙浏览器算子且昏暗和〖K ^ n（X）<∞} ，對於某些 n∈N。我們的方法推廣到巴纳赫空間，艾埃娜為作用於希爾伯特空間 的算子提供了一些結果. 关键词: 擬弗雷德霍尔姆算子，具有拓撲均勻下降的算子，有限範圍冪的算子

I. I

In the sequel, 𝑋 will be denote an infnite

dimensional normed space and𝐿(𝑋) denote the

algebra of all bounded linear operators acting on 𝑋 . For 𝑇 ∈ 𝐿(𝑋) , we denote by 𝑁(𝑇) the null space of 𝑇, 𝑅(𝑇) = 𝑇(𝑋) the range of 𝑇,𝛼(𝑇) = 𝑑𝑖𝑚 𝑁(𝑇) the nullity of 𝑇 and by 𝛽(𝑇) = 𝑐𝑜𝑑𝑖𝑚 𝑅(𝑇) = dim 𝑋 = 𝑅(𝑇) the defect of T. Given 𝑛 ∈ ℕ , we denote by 𝑇𝑛 the restriction

of𝑇 ∈ 𝐿(𝑋) onthe subspace 𝑅(𝑇𝑛) = 𝑇𝑛(𝑋).

The principal research method used in this work is the hypothetical-deductive method. In this work we employed a particular notion of topological descent, called topological descent

uniform (abbreviated TDU), to find a

descomposition property (similar to type Kato descomposition) for certain clases of operator. By using this descomposition, we stablished and proved some results which provide a moregeneral framework to able to extent results given in Hilbert spaces to a Banach spaces and we obtain interesting applications of these results for perturbation of some clases of operators.

The main results of our research are the following. We give new results on the stability of operators with topological finite descent under perturbations by operators with finite-range power and we characterized the class of semi B-Weyl operator as the following form: 𝑇 is a lower semi B-Weyl operator if and only if 𝑇 = 𝑆 + 𝐾 where 𝐾 is a lower semi B-Browder operator and

dim 𝐾𝑛(𝑋) < ∞, for some 𝑛 ∈ ℕ.

The results of our research can be potentially important for mathematicians who working in áreas such that operators theory, perturbation theory and others afine areas.

This generalizes the corresponding result of Aiena for operators on Hilbert spaces. Although the above theorems have been showed under the hypothesis that X is a Banach space ([1, Theorem 12.7] and [1, Theorem 21.3], respectively). We can see that the conclusions of these theorems remain true if the hypothesis Banach is replaced by X is a normed space. The following Theorem was proved in [1, Theorem 21.9] under the hypothesis that S compact or quasinilpotent operator. However, we can observe that this conclusion remain true if we suppose that the operator S has a finite-range power

The following Lemma was proved in [2] under the hypothesis that T is a quasiFredholm operator. However, we can observe that this conclusion remain true if we suppose that the operator T has TUD.

In the following theorem, we construct a decomposition property for operators that have TUD, similar to the Kato type operator.

As in Lemma 2.8, the condition T is a quasi-Fredholm operator can be replaced by the hypothesis T has TUD, of Theorem [2, Theorem 2.8], and the conclusion of its remains true.

The following theorem extends the results obtained in [3, Theoren 2.8] to a larger class of operators.

The following theorem was established in [4] for the particular case of Hilbert spaces. Now, we extend this to the context of Banach spaces.

II. P

Definition 1.1: Let 𝑋 be a normed space and 𝑇 ∈ 𝐿(𝑋), then

(i)𝑇 is said to be semi-regular if 𝑅(𝑇) is

closed and 𝑁(𝑇) ⊆ 𝑅(𝑇𝑛_{) for every}

𝑛 ∈ ℕ.

(ii)𝑇 is said to be essentially semi-regular

if 𝑅(𝑇) is closed and there exists a finite

dimensional subspace 𝐹 such that 𝑁(𝑇) ⊆

𝑅(𝑇𝑛_{) + 𝐹 for every𝑛 ∈ ℕ.}

Theorem 1.2: Let 𝑋 be a normed space and

𝑇 ∈ 𝐿(𝑋) . The following conditionsare

equivalent:

(i)𝑇 is semi-regular,

(ii)𝑇𝑛 _{is semi-regular for all 𝑛 ∈ ℕ ,}

(iii)𝑇𝑛 is semi-regular for some 𝑛 ∈ ℕ. Definition 1.3: Let 𝑋be a normed space. An operator 𝑇 ∈ 𝐿(𝑋) is said to beof Kato type if there exists a pair of 𝑇-invariant closed subspaces

(𝑀, 𝑁) suchthat 𝑋 = 𝑀 ⊕ 𝑁, the restriction 𝑇⎸𝑀

is semi regular and 𝑇⎸𝑁 is nilpotent.

Theorem 1.4: Let 𝑋 be a normed space. If 𝑇 ∈ 𝐿(𝑋) is essentially semi-regular then there

exist a decomposition 𝑋 = 𝑋2⊕ 𝑋2 with the

properties that 𝑇(𝑋1) ⊆ 𝑋1 , 𝑇(𝑋2) ⊆ 𝑋2 ,

dim 𝑋1< ∞ , 𝑇⎸𝑋₁ is nilpotent and 𝑇⎸𝑋₂ is semiregular.

Although the above theorems have been showed under the hypothesis that𝑋 is a Banach space ([1,

Theorem 12.7] and [1, Theorem 21.3],

respectively).We can see that the conclusions of

these theorems remain true if the

hypothesisBanach is replaced by 𝑋 is a normed space.The following Theorem was proved in [1,

Theorem 21.9] under the hypothesisthat

𝑆 compact or quasinilpotent operator. However, we can observe that thisconclusion remain true if we suppose that the operator S has a finite-range power.

Theorem 1.5: Let 𝑋 be a Banach space and

let 𝑇 ∈ 𝐿(𝑋) such that T isessentially

semi-regular, dim 𝑆𝑛_{(𝑋) for some 𝑛 ∈ ℕ and}

𝑇𝑆 = 𝑆𝑇, then

III. Q

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For 𝑇 ∈ 𝐿(𝑋) is defined

𝜅𝑛(𝑇) =

𝑑𝑖𝑚 (𝑅(𝑇𝑛_{) ∩ 𝑁(𝑇))}

𝑑𝑖𝑚 (𝑅(𝑇𝑛+1_{) ∩ 𝑁(𝑇))}

In virtue of [1, Corollary 12.23], we have

𝜅𝑛(𝑇) =

𝑑𝑖𝑚 (𝑅(𝑇) + 𝑁(𝑇𝑛))

𝑑𝑖𝑚 (𝑅(𝑇) + 𝑁(𝑇𝑛+1₎₎

Defnition 2.1: Let 𝑇 ∈ 𝐿(𝑋) . If there is

𝑑 ∈ ℕsuch that 𝜅𝑛(𝑇) = 0for𝑛 ≥ 𝑑 , then 𝑇 is

said to have uniform descent for 𝑛 ≥ 𝑑 . If in addition,𝑅(𝑇𝑛) is closed in the operator range topology of 𝑅(𝑇𝑑)for 𝑛 ≥ 𝑑 , then we say that 𝑇 has topological uniform descent (TUD for brevity) for 𝑛 ≥ 𝑑.

Definition 2.2: Let 𝑋 be a Banach space. An

operator 𝑇 ∈ 𝐿(𝑋) is said to bequasi-Fredholm

of degree 𝑑, and is denoted 𝑇 ∈ 𝑄𝐹(𝑑), if there

exists 𝑑 ∈ ℕsuchthat 𝜅𝑛(𝑇) = 0 for each 𝑛 ≥ 𝑑

and 𝑅(𝑇𝑑+1_{) is closed.}

An operator is Fredholm if it is quasi-Fredholm of some degree 𝑑.

Theorem 2.3: [1, Lemma 22.17] Let 𝑋 be a

Banach space and 𝑇 ∈ 𝐿(𝑋) . If 𝑑 ∈ ℕ and

𝜅𝑛(𝑇) < ∞ for each 𝑛 ≥ 𝑑 , then following

statements are equivalent:

(i)There exists 𝑛 ≥ 𝑑 + 1 such that 𝑅(𝑇𝑛_{) is} closed,

(ii)𝑅(𝑇𝑑+1_{) is closed,}

(iii)𝑅(𝑇𝑛) is closed for each 𝑛 ≥ 𝑑,

(iv)𝑅(𝑇𝑖) + 𝑁(𝑇𝑗) is closed for all 𝑖, 𝑗 with 𝑖 + 𝑗 ≥ 𝑑.

From this Theorem we see easily that if 𝑇 is a quasi-Fredholm operator of somedegree 𝑑 then 𝑇

has topological uniform descent for 𝑛 ≥ 𝑑. But

the converse isnot always true as shown in [5]. Theorem 2.4: [6, Theorem 3.2] Let 𝑋 be a

Banach space. If 𝑇 is a boundedoperator with

uniform descent for 𝑛 ≥ 𝑑 on the Banach space 𝑋, the followingare equivalent:

(i)𝑇 has topological uniform descent for

𝑛 ≥ 𝑑.

(ii) There is an 𝑛 ≥ 𝑑 and a positive integer 𝑘

for which 𝑅(𝑇𝑛_{) + 𝑁(𝑇}𝑘_{) isclosed in 𝑋.}

Theorem 2.5: [2, Theorem 2.5] 𝑇 ∈ 𝑄𝐹(𝑑), if

and only if 𝑇∗_{∈ 𝑄𝐹(𝑑).}

Theorem 2.6: [7, Theorem 2] An operator 𝑇 ∈ 𝐿(𝑋) is quasi-Fredholm if andonly if exists

𝑑 ∈ ℕsuch that 𝑅(𝑇𝑑) is closed and 𝑇𝑑is semi regular.

Theorem 2.7: An operator 𝑇 ∈ 𝐿(𝑋) has TUD if and only if exists 𝑑 ∈ ℕ suchthat 𝑇𝑑 is semi regular.

Proof.𝑇𝑑 is semi regular if and only if

𝑁(𝑇𝑑) = 𝑁(𝑇) ∩ 𝑅(𝑇𝑑) ⊆ 𝑅((𝑇𝑑)𝑛) , forall

𝑛 ∈ ℕ and 𝑅(𝑇𝑑) = 𝑅(𝑇𝑑+1) is closed in

𝑅(𝑇𝑑) , or equivalent 𝑁(𝑇) ∩ 𝑅(𝑇𝑑) ⊆ 𝑅(𝑇𝑛) , for all 𝑛 ≥ 𝑑 and 𝑅(𝑇𝑑+1_{) is closed in 𝑅(𝑇}𝑑_), i.e 𝑇 has topological uniformdescent.

The following Lemma was proved in [2] under the hypothesis that 𝑇 is a quasi-Fredholm operator. However, we can observe that this conclusion remain true ifwe suppose that the operator T has TUD.

Lemma 2.8: If 𝑇 ∈ 𝐿(𝑋) and T have TUD then there exists asequence {𝑁𝑗}_𝑗=0

∞

of subspaces of X such that for all 𝑗 = 0,1, 2 … we have

(i)𝑁𝑗⊆ 𝑁𝑗+1,

(ii)𝑇(𝑁𝑗) ⊆ 𝑁𝑗,

(iii)𝑁𝑗⊆ 𝑁(𝑇𝑗),

(iv)𝑁𝑗∩ 𝑅(𝑇𝑑) = 0,

(v)𝑁𝑗+ 𝑅(𝑇𝑑) = 𝑁(𝑇𝑗) + 𝑅(𝑇𝑑)

Lemma 2.9: If 𝑇 ∈ 𝐿(𝑋) and T have TUD then there exists a sequence {𝑀𝑗}_𝑗=0∞ of subspaces of X such that for all 𝑗 = 0,1, 2 … we have

(i)𝑀𝑗+1⊆ 𝑀𝑗, (ii)𝑇(𝑀𝑗) ⊆ 𝑀𝑗, (iii)𝑅(𝑇𝑗_{) ⊆ 𝑀} 𝑗, (iv)𝑀𝑗+ 𝑁(𝑇𝑑) = 𝑋, (v)𝑀𝑗+ 𝑁(𝑇) = 𝑇−1(𝑀𝑗+1), (vi)𝑀𝑗∩ (𝑅(𝑇𝑗) + 𝑁(𝑇𝑑)) ⊆ 𝑅(𝑇𝑗), (vii)𝑀𝑗∩ 𝑁(𝑇𝑑) = 𝑅(𝑇𝑗) ∩ 𝑁(𝑇𝑑).

Proof. Let L be a subspace of X such that

(𝑅(𝑇) + 𝑁(𝑇𝑑)) ⊕ 𝐿 = 𝑋 and we define

𝑀𝑗= {

𝑋, 𝑖𝑓 𝑗 = 0 𝑇(𝑀𝑗−1) + 𝐿, 𝑖𝑓 𝑗 ≥ 1.

Note that 𝑀𝑗 is clearly a subspace of X, for each j.

(i) If 𝑗 = 0, we have 𝑀1= 𝑇(𝑋) + 𝐿 ⊆ 𝑋 =

𝑀0 . Suppose that 𝑀𝑗+1⊆ 𝑀𝑗 ,then 𝑀𝑗+2=

𝑇(𝑀𝑗+1) + 𝐿 ⊆ 𝑇(𝑀𝑗) + 𝐿 = 𝑀𝑗+1 . Therefore,

𝑀𝑗+1⊆ 𝑀𝑗foreach j.

(ii) If 𝑗 = 0, we have 𝑇(𝑀0) = 𝑇(𝑋) ⊆ 𝑋 =

𝑀0. Suppose that 𝑇(𝑀𝑗) ⊆ 𝑀𝑗,then 𝑇(𝑀𝑗+1) =

𝑇(𝑇(𝑀𝑗) + 𝐿) ⊆ 𝑇(𝑀𝑗+ 𝐿) = 𝑇(𝑀𝑗) ⊆

𝑇(𝑀𝑗) + 𝐿 = 𝑇(𝑀𝑗+1) .Therefore, 𝑇(𝑀𝑗) ⊆ 𝑀𝑗

(iii) If 𝑗 = 0 , we have 𝑅(𝑇0) = 𝑋 = 𝑀₀.

Suppose that 𝑅(𝑇𝑗) ⊆ 𝑀𝑗, then

𝑅(𝑇𝑗+1_{) ⊆ 𝑇 (𝑅(𝑇}𝑗_{)) + 𝐿 ⊆ 𝑇(𝑀}

𝑗) + 𝐿 = 𝑀𝑗+1

Consequently, we obtain that 𝑅(𝑇𝑗+1_{) ⊆}

𝑀𝑗+1, and hence 𝑅(𝑇𝑗) ⊆ 𝑀𝑗for eachj.

(iv)If 𝑗 = 0 , we have 𝑀0+ 𝑁(𝑇𝑑) = 𝑋 + 𝑁(𝑇𝑑) = 𝑋 . Suppose that 𝑀𝑗+ 𝑁(𝑇𝑑) = 𝑋 , then 𝑀𝑗+1+ 𝑁(𝑇𝑑) = 𝑇(𝑀𝑗) + 𝐿 + 𝑁(𝑇𝑑) = 𝑇(𝑀𝑗) + 𝑇 (𝑁(𝑇𝑑)) + 𝐿 + 𝑁(𝑇𝑑) = 𝑇(𝑀𝑗+ 𝑁(𝑇𝑑)) + 𝐿 + 𝑁(𝑇𝑑) = 𝑅(𝑇) + 𝑁(𝑇𝑑) + 𝐿 = 𝑋.

Therefore 𝑀𝑗+ 𝑁(𝑇𝑑) = 𝑋 for each j.

(v)Suppose that 𝑥 ∈ 𝑇−1_{(𝑀𝑗+1) , then}

𝑇𝑥 ∈ 𝑀𝑗+1= 𝑇(𝑀𝑗) + 𝐿 and so 𝑇𝑥 = 𝑇𝑚𝑗+ 𝑙 ,

where𝑚𝑗∈ 𝑀𝑗 and 𝑙 ∈ 𝐿. In consequence𝑇(𝑥 −

𝑚𝑗) = 𝑙 , thus 𝑥 − 𝑚𝑗∈ 𝑁(𝑇) because 𝐿 ∩

𝑅(𝑇) = 0 . Hence 𝑥 ∈ 𝑁(𝑇) + 𝑀𝑗 .

Therefore𝑇−1_(𝑀

𝑗+1) ⊆ 𝑀𝑗+ 𝑁(𝑇).

On the other hand, if 𝑥 ∈ 𝑀𝑗+ 𝑁(𝑇) then

𝑇𝑥 ∈ 𝑇(𝑀𝑗) and so 𝑇𝑥 ∈ 𝑇(𝑀𝑗) + 𝐿 = 𝑀𝑗+1. In

consequence 𝑥 ∈ 𝑇−1(𝑀𝑗+1).

(vi) If 𝑗 = 0 , we have 𝑀0∩ (𝑅(𝑇0) +

𝑁(𝑇𝑑)) = 𝑋 ∩ (𝑋 + 𝑁(𝑇𝑑)) = 𝑋 = 𝑅(𝑇0).

Now suppose that 𝑀𝑗∩ (𝑅(𝑇𝑗) + 𝑁(𝑇𝑑)) ⊆

𝑅(𝑇𝑗_). Observe that 𝑀𝑗+1∩ (𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑)) = 𝑀𝑗+1∩ (𝑅(𝑇𝑗+1_{) + 𝑁(𝑇}𝑑_{)) ∩ 𝑀} 1∩ (𝑅(𝑇) + 𝑁(𝑇𝑑)) = 𝑀𝑗+1∩ (𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑)) ∩ (𝑅(𝑇) + 𝐿) ∩ (𝑅(𝑇) + 𝑁(𝑇𝑑_{)) = 𝑀} 𝑗+1∩ (𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑)) ∩ 𝑅(𝑇) = 𝑀𝑗+1∩ (𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑) ∩ 𝑅(𝑇)), also 𝑇 (𝑅(𝑇𝑗) + 𝑁(𝑇𝑑+1)) = 𝑇 (𝑅(𝑇𝑗)) + 𝑇 (𝑁(𝑇𝑑+1)) = 𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑) ∩ 𝑅(𝑇). So 𝑇−1(𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑) ∩ 𝑅(𝑇)) = 𝑅(𝑇𝑗) + 𝑁(𝑇𝑑+1) + 𝑁(𝑇) = 𝑅(𝑇𝑗) + 𝑁(𝑇𝑑). Then if 𝑥 ∈ 𝑀𝑗+1∩ (𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑)) = 𝑀𝑗+1∩ (𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑) ∩ 𝑅(𝑇)) we have 𝑇−1({𝑥}) ⊆ 𝑇−1(𝑀𝑗+1) ∩ 𝑇−1(𝑅(𝑇𝑗+1) + 𝑁(𝑇𝑑)) . In consequence 𝑇−1_{({𝑥}) ⊆ (𝑀} 𝑗+ 𝑁(𝑇)) ∩ (𝑅(𝑇𝑗) + 𝑁(𝑇𝑑)). Thus 𝑇−1_{({𝑥}) ⊆ 𝑀} 𝑗∩ (𝑅(𝑇𝑗) + 𝑁(𝑇𝑑)), hence 𝑇−1({𝑥}) ⊆ (𝑅(𝑇𝑗)) and so 𝑇(𝑇−1_{({𝑥})) ⊆ 𝑇(𝑅(𝑇}𝑗_{). Therefore𝑥 ∈ 𝑅(𝑇}𝑗+1_).

(vii) It follows of (iii) and (vi).

In the following theorem, we construct a decomposition property for operators that have TUD, similar to the Kato type operator.

Theorem 2.10: If𝑇 ∈ 𝐿(𝑋) and T have TUD, then there exist T invariantsubspaces M and N of

X such that, 𝑋 = 𝑀 ⊕ 𝑁,𝑇⎸𝑀 is semi regular and 𝑇⎸𝑁 is nilpotent.

Proof. From Lemma 2.8 and 2.9 it follows that there are sequences of subspacesof X, {𝑁𝑗}_𝑗=0

∞

and {𝑀𝑗}_𝑗=0 ∞

that satisfy certain

conditions. Let 𝑀 = 𝑀𝑑 and𝑁 = 𝑁𝑑, then of (iv)

and (iii) of Lemma 2.9, and (v) of Lemma 2.8 it follows that

𝑀 + 𝑁 = 𝑀 + 𝑅(𝑇𝑑) + 𝑁

= 𝑀 + 𝑅(𝑇𝑑_{) + 𝑁(𝑇}𝑑_{) = 𝑋}

Also, of (iii) and (iv) of Lemma 2.8, and (v) of Lemma 2.9 it follows that

𝑀 ∩ 𝑁 = 𝑀 ∩ 𝑁 ∩ 𝑁(𝑇𝑑) = 𝑅(𝑇𝑑) ∩ 𝑁(𝑇𝑑) ∩

𝑁 = {0},

so, 𝑋 = 𝑀 ⊕ 𝑁. On the other hand, since T have TUD follows that

𝑁[(𝑇⎸𝑀)𝑑] = 𝑀 ∩ 𝑁(𝑇𝑑) = 𝑅(𝑇𝑑) ∩ 𝑁(𝑇𝑑) =

𝑅(𝑇𝑑+𝑗) ∩ 𝑁(𝑇𝑑) ⊆ 𝑅(𝑇𝑑+𝑗),

for all 𝑗 = 0,1, 2 … . Also, 𝑇𝑑_{(𝑋) = 𝑇}𝑑_{(𝑀 ⊕}

𝑁) = 𝑇𝑑_{(𝑀) = 𝑅[(𝑇⎸}

𝑀)𝑑] it follows that

𝑅[(𝑇⎸𝑀)𝑑]is closed. This shows that (𝑇⎸𝑀)𝑑is

semi-regular.Hence, by Theorem 1.2, 𝑇⎸𝑀 is

semi-regular.

Theorem 2.11: Let X a Banach space, 𝑇 ∈ 𝐿(𝑋)and suppose that there existT invariant subspaces M and N of X such that 𝑋 = 𝑀 ⊕ 𝑁,

𝑇⎸𝑀 is essentiallysemi-regular and 𝑇⎸𝑁 is nilpotent, then T have TUD.

Proof. Suppose that there exist T invariant subspaces M and N of X such that 𝑋 = 𝑀 ⊕ 𝑁,

𝑇⎸𝑀 is essentially semi-regular and 𝑇⎸𝑁 is

nilpotent.Let 𝑀1 and 𝑁1 subspaces T invariant of

Msuch that 𝑇⎸𝑀1is semi-regular, 𝑇⎸𝑁1is nilpotent

and 𝑀 = 𝑀1⊕ 𝑁1 (see Theorem 1.4). If 𝑑 ∈

ℕsuch that (𝑇⎸𝑁)𝑑_{= 0and (𝑇⎸𝑁}

1)

𝑑

= 0, then

𝑅(𝑇𝑛) = 𝑇𝑛(𝑋) = 𝑇𝑛((𝑀1⊕ 𝑁1) ⊕ 𝑁)

= 𝑇𝑛(𝑀1)

for all 𝑛 ≥ 𝑑, so 𝑅(𝑇𝑛) is closed for all 𝑛 ≥ 𝑑. Also,

𝑁(𝑇) ∩ 𝑅(𝑇𝑑_{) = 𝑁(𝑇) ∩ 𝑇}𝑑_(𝑀

1) ⊆ 𝑁(𝑇) ∩ 𝑀1

⊆ 𝑇𝑛_(𝑀

1) = 𝑅(𝑇𝑛)

for all 𝑛 ≥ 𝑑, and consequently 𝑇𝑑is semi-regular.

By Theorem 2.6, we concludethat T is quasi-Fredholm.

Corollary 2.12: If X a Banach space and 𝑇 ∈ 𝐿(𝑋) is of Kato type, then T have TUD.

As in Lemma 2.8, the condition T is a quasi-Fredholm operator can be replacedby the hypothesis T has TUD, of Theorem [2, Theorem 2.8], and the conclusión of its remains true.

Theorem 2.13: If 𝑇 ∈ 𝐿(𝑋) has topological

uniform descent for 𝑛 ≥ 𝑑 and𝑁(𝑇𝑑) + 𝑅(𝑇) is

complemented, then there exist T invariant subspaces M andN of X such that, Mis closed,𝑋 =

𝑀 ⊕ 𝑁,𝑇⎸𝑀is semi-regular and 𝑇⎸𝑁is nilpotent.

Corollary 2.14: If 𝑇 ∈ 𝐿(𝑋) has topological

uniform descent for 𝑛 ≥ 𝑑 and𝑁(𝑇𝑑_{) + 𝑅(𝑇) is}

complemented, then T is quasi-Fredholm.

The following theorem extends the results obtained in [3, Theoren 2.8] to alarger class of operators.

Theorem 2.15: If 𝑇 ∈ 𝑄𝐹(𝑑), 𝑁(𝑇𝑑) + 𝑅(𝑇)

is complemented, 𝑆 ∈ 𝐿(𝑋) suchthat

dim [𝑆𝑛(𝑋)] < ∞ for some 𝑛 ∈ ℕ and TS = ST,

then T + S is quasiFredholm.

Proof. Suppose that T is a operator quasi-Fredholm. By Theorem [2, Theorem2.8], it follows that there exist T invariant subspaces M

and N of X such that Mis closed, 𝑋 = 𝑀 ⊕

𝑁,𝑇⎸𝑀 is semi-regular and 𝑇⎸𝑁 is nilpotent. Let

𝑝, 𝑞 ∈ ℕ be such that dim [𝑆𝑝(𝑋)] < ∞ and

(𝑇⎸𝑁)𝑞= 0, then (𝑇 + 𝑆)𝑝+𝑞= ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝+𝑞 𝑘=0 𝑆𝑘 = ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=0 𝑆𝑘 + ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝+𝑞 𝑘=𝑝+1 𝑆𝑘 = 𝑇𝑝+𝑞_{+ ∑ 𝑐} 𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘 + ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝+𝑞 𝑘=𝑝+1 𝑆𝑘 = 𝑇𝑝+𝑞 + 𝑆(∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘−1₎ + 𝑆𝑝( ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝+𝑞 𝑘=𝑝+1 𝑆𝑘−𝑝) Also, 𝑆 (∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘−1) (𝑀) = ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘(𝑀) ⊆ ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=𝑝 (𝑋) = ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 (𝑀 + 𝑁) = ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 (𝑀) ⊆ 𝑀 𝑆 (∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘−1) (𝑁) = ∑ 𝑐𝑝+𝑞,𝑘𝑆𝑘(𝑇𝑝+𝑞−𝑘(𝑁) 𝑝 𝑘=1 ) = 0 𝑇𝑝+𝑞(𝑀) ⊆ 𝑀 and 𝑇𝑝+𝑞(𝑁) = 0 . Then 𝑇𝑝+𝑞_∈

𝐿(𝑀) is semi-regular (see Theorem 1.2) and 𝑆(∑𝑝𝑘=1𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘𝑆𝑘−1) ∈ 𝐿(𝑀) has finite-range power, from Theorem 1.5 follows

that𝑇𝑝+𝑞+ 𝑆(∑𝑝𝑘=1𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘𝑆𝑘−1) ∈ 𝐿(𝑀)

𝑇𝑝+𝑞+ 𝑆 (∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝

𝑘=1

𝑆𝑘−1) (𝑁) = 0

From Theorem 2.11 that

𝑇𝑝+𝑞_{+ 𝑆 (∑ 𝑐} 𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘−1₎ is quasi-Fredholm, in consequence (𝑇 + 𝑆)𝑝+𝑞 _{= 𝑇}𝑝+𝑞_{+ ∑ 𝑐} 𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝 𝑘=1 𝑆𝑘 + ∑ 𝑐𝑝+𝑞,𝑘𝑇𝑝+𝑞−𝑘 𝑝+𝑞 𝑘=𝑝+1 𝑆𝑘

is quasi-Fredholm (see [8, Theorem 15]). Thus, we can conclude that T + S isquasi-Fredholm (see Theorem 1.2).

Corollary 2.16: If 𝑇 ∈ 𝐿(𝑋), has topological

uniform descent for 𝑛 ≥ 𝑑 , 𝑁(𝑇𝑑_{) + 𝑅(𝑇) is}

complemented, 𝑆 ∈ 𝐿(𝑋) such that

dim [𝑆𝑛(𝑋)] < ∞ for some 𝑛 ∈ ℕ and TS = ST,

then T + S has topological uniform descent. By duality (see Theorem 2.5), we have. Corollary 2.17: If 𝑇 ∈ 𝑄𝐹(𝑑) , 𝑁(𝑇) ∩

𝑅(𝑇𝑑) is complemented, 𝑆 ∈ 𝐿(𝑋) such that

dim [𝑆𝑛(𝑋)] < ∞ for some 𝑛 ∈ ℕ and TS = ST,

then T + S is quasiFredholm.

Similar to Theorem 2.15, we have the following result.

Theorem 2.18: If X is a Banach space, 𝑇 ∈ 𝐿(𝑋), is a Kato type operator, and𝑆 ∈ 𝐿(𝑋), such that dim [𝑆𝑛_{(𝑋)] < ∞ , for some 𝑛 ∈ ℕ} and TS = ST, then T + S is of Kato type operator.

IV. N

C

C

O

L

S

B-W

O

According [5] and [9], T is said to be semi Fredholm (resp. Fredholm,upper semi B-Fredholm, lower semi B-Fredholm), if for some

integer 𝑛 ≥ 0 therange 𝑅(𝑇𝑛) is closed and 𝑇_𝑛,

viewed as an operator from the space 𝑅(𝑇𝑛_{) into}

itself, is a semi-Fredholm (respectively,

Fredholm, upper semi-Fredholm,

lowersemi-Fredholm) operator. Analogously, 𝑇 ∈ 𝐿(𝑋) is

said to be Browder (resp.upper semi B-Browder, lower semi B-B-Browder, B-Weyl, upper semi B-Weyl,lower semi B-Weyl), if for some

integer 𝑛 ≥ 0 the range 𝑅(𝑇𝑛_{) is closed and 𝑇}

𝑛is a Browder (resp. upper semi-Browder, lower semi -Browder, Weyl, upper semi B-Weyl, lower semi B-Weyl) operator.

Theorem 3.1: Let X be a Banach space and

let 𝑇 ∈ 𝑄𝐹(𝑑). If 𝑁(𝑇𝑑_{) + 𝑅(𝑇) is a}

complemented subspace of X, then T is upper semi B-Weyl operator if only if there exist T invariant closed subspacesM and N of X such

that 𝑋 = 𝑀 ⊕ 𝑁 , 𝑇⎸𝑀 is upper semi Weyl

and𝑇𝑛_{(𝑁) = {0} for some 𝑛 ∈ ℕ.}

Proof. Suppose that 𝑇 ∈ 𝑄𝐹(𝑑) is an upper

semi B-Weyl, then 𝑁(𝑇) ∩ 𝑅(𝑇𝑑) is

complemented. Since that 𝑁(𝑇𝑑) + 𝑅(𝑇) is

complemented, From [2, Theorem 2.8 and Lemma 2.7] it follows that there exist T invariant closed subspaces M and N of Xsuch that:

(i)𝑅(𝑇𝑑_{) is a subspace of M,} (ii)𝑋 = 𝑀 ⊕ 𝑁, (iii)𝑇⎸𝑀 is semi-regular, (iv)𝑇𝑑_{(𝑁) = 0.} Then, 𝑇𝑛(𝑋) = 𝑇𝑛_{(𝑀 ⊕ 𝑁) = 𝑇}𝑛_{(𝑀) = 𝑅[(𝑇⎸} 𝑀)𝑛] , for all 𝑛 ≥ 𝑑.

Now, by [1, Corollary 12.23] and since 𝑇⎸𝑀 is semi-regular, thus we have

𝑅(𝑇𝑑₎ 𝑅(𝑇𝑑+1₎= 𝑅[(𝑇⎸𝑀)𝑑] 𝑅[(𝑇⎸𝑀)𝑑+1] = 𝑀 𝑅(𝑇⎸𝑀) + 𝑁[(𝑇⎸𝑀)𝑑] = 𝑀 𝑅(𝑇⎸𝑀) So, 𝛽(𝑇⎸𝑀) = dim [ 𝑀 𝑅(𝑇⎸_𝑀) ] = 𝑑𝑖𝑚 [_𝑅(𝑇𝑅(𝑇𝑑+1𝑑)₎] = 𝑑𝑖𝑚 [ 𝑅(𝑇𝑛) 𝑅(𝑇𝑛+1₎] = 𝛽(𝑇𝑛), for all 𝑛 ≥ 𝑑.Also, 𝛼(𝑇⎸𝑀) = dim[𝑁(𝑇) ∩ 𝑀] = 𝑑𝑖𝑚[𝑁(𝑇) ∩ 𝑁(𝑇𝑑) ∩ 𝑀] = 𝑑𝑖𝑚[𝑁(𝑇) ∩ 𝑁(𝑇𝑑₎ ∩ 𝑅(𝑇𝑑)] = 𝑑𝑖𝑚[𝑁(𝑇) ∩ 𝑅(𝑇𝑑)] = 𝑑𝑖𝑚[𝑁(𝑇) ∩ 𝑅(𝑇𝑛)] = 𝛼(𝑇_𝑛) < ∞ For all 𝑛 ≥ 𝑑. Therefore

𝛼(𝑇⎸𝑀) − 𝛽(𝑇⎸𝑀) = 𝛼(𝑇𝑛) − 𝛽(𝑇𝑛) ≤ 0

Conversely, suppose that there there exist T invariant closed subspaces M and N of X such

that 𝑋 = 𝑀 ⊕ 𝑁, 𝑇⎸𝑀 is upper semi Weyl and

𝑇𝑛(𝑁) = {0} for some 𝑛 ∈ ℕ. Then,

𝑅(𝑇𝑛) = 𝑇𝑛_{(𝑋) = 𝑇}𝑛_{(𝑀 ⊕ 𝑁) = 𝑇}𝑛_{(𝑀) =}

𝑅[(𝑇⎸𝑀)𝑛], for all 𝑛 ≥ 𝑑. It follows that 𝑅(𝑇𝑛) is closed for all 𝑛 ≥ 𝑑 , and 𝑇𝑑= 𝑇⎸𝑅(𝑇𝑑) = 𝑇⎸_{𝑅[(𝑇⎸}

𝑀)𝑑] = (𝑇⎸𝑀)𝑑.

This shows that T is upper semi B-Weyl.

The following theorem was established in [4] for the particular case of Hilbert spaces. Now, we extend this to the context of Banach spaces.

Theorem 3.2: Let X be a Banach space and 𝑇 ∈ 𝐿(𝑋). Then T is lower semiB-Weyl, if and only if there exist 𝑆, 𝐾 ∈ 𝐿(𝑋), such that T = S +

K, S is lowersemi B-Browder and

dim [𝐾𝑛_{(𝑋)] < ∞, for some 𝑛 ∈ ℕ.}

Proof.(⇒) This is an immediate consequence of [2, Theorem 3.4].

(⇐)Conversely, suppose that there exist 𝑆, 𝐾 ∈ 𝐿(𝑋) such that T = S + K, S is lower semi

B-Browder and dim [𝐾𝑛(𝑋)] < ∞ , for some

𝑛 ∈ ℕ. By [3, Theoren 2.8] T is lowersemi B-Browder, in consequence T is lower semi B-Weyl.

R

[1] MÜLLER, V. (2007). Spectral theory of

linear operators and spectral systems in

Banach algebras. Operator Theory: Advances

and Applications, 139, Basel: Birkhaüser,

Springer. doi: 10.1007/978-3-0348-7788-6

[2] GARCÍA,

O.

CAUSIL,

D.

CARPINTERO, C. and SANABRIA, J.

(2020). New decompositions for the classes

of

quasi-Fredholm

and

semi

B-Weyl

operators. Linear and Multilinear Algebra,

68(4),

pp.

750-763.

doi:

10.1080/03081087.2018.1517718

[3] AIENA, P. and TRIOLO, S. (2016).

Some perturbation results through localized

SVEP. Acta Scientiarum Mathematicarum,

82(12), pp. 205-219. doi:

10.14232/actasm-014-785-1

[4] AIENA, P. and SANABRIA, J.(2008).

On left and right poles of the resolvet. Acta

Scientiarum Mathematicarum, 74, pp.

669-687.

[5] BERKANI, M. (2000). Restriction of an

operator to the range of its powers. Studia

Mathematica, 140(2), pp. 163-175.

[6] GRABINER, S. (1982). Uniform ascent

and descent of bounded operators. Journal of

the Mathemativcal Society of Japan, 34(2),

pp. 317-337.

[7] CARPINTERO , C., GARCÍA, O.,

ROSAS, E. and SANABRIA, J. (2008).

B-Browder Spectra an Localized SVEP.

Rendiconti del Circolo Matematico di

Palermo,

57(2),

pp.

241-255.

doi:

10.1007/s12215-008-0017-4

[8] MBEKHTA,

M.

and

MÜLLER,

V.(1996). On the axiomatic theory of

spectrum II. Studia Mathematica, 119(2), pp.

129-147.

[9] BERKANI, M. (2001). On semi

B-Fredholm operators. Glasgow Mathematical

Journal, 43, pp. 457-465.

参考文:

MÜLLER，V.（2007）。巴纳赫代數中線

性算子和光譜系統的光譜理論。運算符理

論：高級與應用，139，巴塞爾：比尔库

瑟 ， 施 普 林 格 。 doi ： 10.1007 /

978-3-0348-7788-6

[2]

GARCÍA ， O. CAUSIL ， D.

CARPINTERO ， C. 和 SANABRIA ， J.

（2020）。擬弗雷德霍尔姆和半魏尔算子

類別的新分解。線性和多線性代數，68

（4），第 750-763 頁。土井：10.1080 /

03081087.2018.1517718

[3] AIENA ， P 。 和 TRIOLO ， S 。

（2016）。局部 SVEP 會引起一些干擾。

數 學 學 報 ， 82 （ 12 ） , 第 205-219 頁 。

doi：10.14232 / actasm-014-785-1

[4] AIENA ， P. 和

SANABRIA ， J.

（2008）。在解決方案的左右兩極。數學

學報，74，第 669-687 頁。

[5] BERKANI，M。（2000）。將操作員

限 制 在 其 權 限 範 圍 內 。 數 學 研 究 ， 140

（2），第 163-175 頁。

[6] GRABINER，S.（1982）。有界算子

的均勻上升和下降。日本數學學會雜誌，

34（2），第 317-337 頁。

[7] CARPINTERO ， C. ，GARCÍA ，O. ，

ROSAS，E. 和 SANABRIA，J.（2008）。

乙浏览器光譜是局部 SVEP。巴勒莫圓形

劇場，57（2），第 241-255 頁。 doi：

10.1007 / s12215-008-0017-4

[8] MBEKHTA ， M. 和 MÜLLER ， V.

（1996）。關於頻譜的公理理論 II。數學

研究，119（2），第 129-147 頁。

[9] BERKANI，M。（2001）。在半弗雷

德霍尔姆運算符上。格拉斯哥數學雜誌，

43，第 457-465 頁。