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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# On new strong versions of Browder type theorems

José Sanabria
• Corresponding author
• Departamento de Matemáticas, Universidad de Oriente, Cumaná, Venezuela
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• Other articles by this author:
/ Carlos Carpintero
• Departamento de Matemáticas, Universidad de Oriente, Cumaná, Venezuela
• Vicerrectoría de Investigación, Universidad Autónoma del Caribe, Barranquilla, Colombia
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• Other articles by this author:
/ Jorge Rodríguez
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/ Ennis Rosas
• Universidad de la Costa, Departamento de Ciencias Naturales y Exactas, Barranquilla, Colombia
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• Other articles by this author:
/ Orlando García
Published Online: 2018-04-02 | DOI: https://doi.org/10.1515/math-2018-0029

## Abstract

An operator T acting on a Banach space X satisfies the property (UWΠ) if σa(T)∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T), where σa(T) is the approximate point spectrum of T, $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) is the upper semi-Weyl spectrum of T and Π(T) the set of all poles of T. In this paper we introduce and study two new spectral properties, namely (VΠ) and (VΠa), in connection with Browder type theorems introduced in [1], [2], [3] and [4]. Among other results, we have that T satisfies property (VΠ) if and only if T satisfies property (UWΠ) and σ(T) = σa(T).

MSC 2010: 47A10; 47A11; 47A53

## 1 Introduction and preliminaries

Throughout this paper, L(X) denotes the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space X. We refer to [5] for details about notations and terminologies. However, we give the following notations that will be useful in the sequel:

• Browder spectrum: σb(T)

• Weyl spectrum: σW(T)

• Upper semi-Browder spectrum: σub(T)

• Upper semi-Weyl spectrum: $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T)

• Drazin invertible spectrum: σD (T)

• B-Weyl spectrum: σBW(T)

• Left Drazin invertible spectrum: σLD (T)

• Upper semi B-Weyl spectrum: $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T)

• approximate point spectrum: σa(T)

• surjectivity spectrum: σs(T)

In this paper, we introduce two new spectral properties of Browder type theorems, namely, the properties (VΠ) and (VΠa), respectively. In addition, we establish the precise relationships between these properties and others variants of Browder’s theorem have been recently introduced in [1], [2], [3] and [4].

Denote by iso K, the set of all isolated points of K ⊆ ℂ. If TL(X) define $E0(T)={λ∈isoσ(T):0<α(λI−T)<∞},Ea0(T)={λ∈isoσa(T):0<α(λI−T)<∞},E(T)={λ∈isoσ(T):0<α(λI−T)},Ea(T)={λ∈isoσa(T):0<α(λI−T)}.$

Also, define $Π0(T)=σ(T)∖σb(T),Πa0(T)=σa(T)∖σub(T),Π(T)=σ(T)∖σD(T),Πa(T)=σa(T)∖σLD(T).$

Recall that TL(X) is said to satisfy a-Browder’s theorem (resp., generalized a-Browder’s theorem) if σa(T)∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\left(T\right)={\mathit{\Pi }}_{a}^{0}\end{array}$(T) (resp., σa(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Πa(T)). From [6, Theorem 2.2] (see also [7, Theorem 3.2(ii)]), a-Browder’s theorem and generalized a-Browder’s theorem are equivalent. It is well known that a-Browder’s theorem for T implies Browder’s theorem for T, i.e. σ(T)∖σW(T) = Π0(T). Also by [6, Theorem 2.1], Browder’s theorem for T is equivalent to generalized Browder’s theorem for T, i.e. σ(T) ∖ σBW(T) = Π(T).

Now, we describe several spectral properties introduced recently in [8], [2], [3], [9] and [10].

#### Definition 1.1

An operator TL(X) is said to have:

• property (gab) [8] if σ(T) ∖ σBW(T) = Πa(T).

• property (az) [9] if σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\left(T\right)={\mathit{\Pi }}_{a}^{0}\end{array}$(T).

• property (gaz) [9] if σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Πa(T).

• property (ah) [10] if σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\left(T\right)={\mathit{\Pi }}_{a}^{0}\end{array}$(T).

• property (gah) [10] if σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Πa(T).

• property (Sb) [2] if σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Π0(T).

• property (Sab) [3] if σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\left(T\right)={\mathit{\Pi }}_{a}^{0}\end{array}$ (T).

According to [9, Corollary 3.5], [10, Theorem 2.14] and [10, Corollary 2.15], we have that the properties (az), (gaz), (ah) and (gah) are equivalent. It was proved in [3, Corollary 2.9], that properties (Sb) and (Sab) are equivalent.

## 2 Properties (VΠ) and (VΠa)

According to [1], TL(X) has property (WΠ) (resp. property (UWΠa)) if σ(T) ∖ σW(T) = Π(T) (resp. σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T)). It was shown in [1, Theorem 2.4] (resp. [1, Theorem 2.2]) that property (WΠ) (resp. (UWΠa)) implies generalized Browder’s theorem (resp. property (WΠ)) but not conversely. Following [1], an operator TL(X) is said to have property (UWΠ) if σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T). It was shown in [1, Theorem 3.5] that property (UWΠ) implies property (WΠ) but not conversely. According to [4], TL(X) has property (ZΠa) if σ(T) ∖ σW(T) = Πa(T). In this section, we introduce and study two equivalent spectral properties that are stronger than the properties (UWΠa), (UWΠ) and (ZΠa).

#### Definition 2.1

An operator TL(X) is said to have property (VΠ) if σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T).

#### Example 2.2

1. Let TL(2(ℕ)) be defined by $T(x1,x2,x3,⋯)=x22,x33,⋯.$

Since σ(T) = $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = {0} and Π(T) = ∅, then σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) and hence T has property (VΠ).

2. Consider the Volterra operator V on the Banach space C[0, 1] defined by V(f)(x) = $\begin{array}{}{\int }_{0}^{x}\end{array}$ f(t)dt for all fC[0, 1]. Note that V is injective and quasinilpotent. Thus, σ(V) = {0} and Π(V) = ∅. Since the range R(V) is not closed, then $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(V) = {0}. Therefore, σ(V) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(V) = Π(V), that means V has property (VΠ).

According to [11], an operator TL(X) is said to have property (VE) if σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = E(T).

The next result gives the relationship between the properties (VE) and (VΠ).

#### Theorem 2.3

For TL(X), the following statements are equivalent:

1. T has property (VE),

2. T has property (VΠ) and E(T) = Π(T).

#### Proof

(i)⇒(ii). Suppose that T satisfies property (VE), i.e. σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = E(T). Since by [11, Theorem 2.10], we have E(T) = Π(T), it follows that T has property (VΠ) and E(T) = Π(T).

(ii)⇒(i). Suppose that T satisfies property (VΠ) and E(T) = Π(T). Then, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) = E(T) and T has property (VE). □

#### Remark 2.4

By Theorem 2.3, property (VE) implies property (VΠ). However, the converse is not true in general. Consider the operator T in Example 2.2, then T satisfies property (VΠ) and as E(T) = {0}, it follows that T does not satisfy property (VE).

#### Theorem 2.5

For TL(X), the following statements are equivalent:

1. T has property (VΠ),

2. T has property (UWΠ) and σ(T) = σa(T),

3. T has property (UWΠa) and σ(T) = σa(T).

#### Proof

(i)⇒(ii). Suppose that T satisfies property (VΠ) and let λ ∈ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T). Since σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ⊆ σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T), we have λ ∈ Π(T) and so, σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ⊆ Π(T).

To show the opposite inclusion Π(T) ⊆ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T), let λ ∈ Π(T). Then, λ ∈ Ea(T), it follows that λ ∈ iso œa(T) and αIT) > 0, so λ ∈ σa(T). As T satisfies property (VΠ) and λ ∈ Π(T), it follows that λ IT is upper semi-Weyl. Therefore, λ ∈ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T). Thus, Π(T) ⊆ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) and T satisfies property (UWΠ). Consequently, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) and σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T). Therefore, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) and σ(T) = σa(T).

(ii)⇒(i). Suppose that T satisfies property (UWΠ) and σ(T) = σa(T). Then, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T). Thus, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) and T satisfies property (VΠ).

(ii)⇔(iii). Obvious. □

The next example shows that, in general, property (UWΠa) does not imply property (VΠ).

#### Example 2.6

Let R be the unilateral right shift operator on ℓ2(ℕ) and UL(2(ℕ)) be defined by $U(x1,x2,x3,⋯)=(0,x2,x3,⋯).$

Define an operator T on X = 2(ℕ) ⊕ 2(ℕ) by T = RU. Then, σ(T) = D(0, 1), the closed unit disc on ℂ, σa(T) = Γ ∪ {0}, where Γ denotes the unit circle ofand $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Γ. Moreover, Πa(T) = {0} and Π(T) = ∅. Therefore, σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T) and σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ≠ Π(T). Thus, T satisfies properties (UWΠa), but T does not satisfy property (VΠ).

The next example shows that, in general, property (UWΠ) does not imply property (VΠ).

#### Example 2.7

Let R be the unilateral right shift operator on ℓ2(ℕ). Define an operator T on X = 2(ℕ) ⊕ 2(ℕ) by T = 0⊕R. Then, σ(T) = D(0, 1), σa(T) = $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Γ∪{0} and Π(T) = ∅. Therefore, σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) and σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ≠ Π(T). Thus, T satisfies property (UWΠ), but T does not satisfy property (VΠ).

The next result gives the relationship between the properties (VΠ) and (WΠ).

#### Theorem 2.8

Let TL(X). Then T has property (VΠ) if and only if T has property (WΠ) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T).

#### Proof

Sufficiency: Suppose that T satisfies property (VΠ), then by Theorem 2.5, T satisfies property (UWΠ). Property (UWΠ) implies by [1, Theorem 3.5] that T satisfies property (WΠ). Consequently, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) and σ(T) ∖ σW(T) = Π(T). Therefore, $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T).

Necessity: Suppose that T satisfies property (WΠ) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T). Then, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ σW(T) = Π(T), and so T satisfies property (VΠ). □

The next example shows that, in general, property (WΠ) does not imply property (VΠ).

#### Example 2.9

Let R be the unilateral right shift operator on ℓ2(ℕ). Since σ(R) = σW(R) = D(0, 1), $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(R) = Γ and Π(R) = ∅, then $σ(R)∖σW(R)=Π(R),σ(R)∖σSF+−(R)≠Π(R).$

Hence, R satisfies property (WΠ), but R does not satisfy property (VΠ).

#### Theorem 2.10

Suppose that TL(X) has property (VΠ). Then:

1. T has property (ZΠa),

2. $\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$(T) = Πa(T) = Π0(T) = Π(T).

#### Proof

1. Property (VΠ) implies by Theorem 2.5 that σ(T) = σa(T), and also implies by Theorem 2.8 that $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T). Hence, σ(T) ∖ σW(T) = σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) = Πa(T) and so T satisfies property (ZΠa).

2. Follows from (i) and [4, Lemma 2.9]. □

The next example shows that, in general, property (ZΠa) does not imply property (VΠ).

#### Example 2.11

Let R be the unilateral right shift operator defined on ℓ2(ℕ). Since σ(R) = σW(R) = D(0, 1), Π(R) = Πa(R) = ∅ and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(R) = Γ, then R satisfies property (ZΠa), but does not satisfy property (VΠ).

#### Remark 2.12

By [4, Lemma 2.9], property (ZΠa) implies that Πa(T) = Π(T). Hence, if TL(X) satisfies property (ZΠa), then σ(T) ∖ σW(T) = Πa(T) = Π(T) and follows that T satisfies property (WΠ). However, the converse is not true in general. For this, consider the operator T in Example 2.7, since σ(T) = σW(T) = D(0, 1), Π(T) = ∅ and Πa(T) = {0}, then T satisfies property (WΠ), but T does not satisfy property (ZΠa).

#### Theorem 2.13

For TL(X), the following statements are equivalent:

1. T has property (VΠ),

2. T has property (gah) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T),

3. T has property (gaz) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T),

4. T has property (ah) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T),

5. T has property (az) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$T).

#### Proof

The equivalences (ii)⇔(iii), (iv)⇔(v) and (ii)⇔(iv) were shown in [10, Corllary 2.15], [10, Theorem 2.14] and [10, Theorem 2.10], respectively.

(i)⇒(ii). Assume that T satisfies property (VΠ). By Theorem 2.5, T satisfies property (UWΠa). Property (UWΠa) implies by [1, Theorem 2.6] that T satisfies generalized a-Browder’s theorem and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Πa$\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$, but by Theorem 2.10, it follow that Πa$\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$ = ∅ and hence, $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T). Consequently, Π(T) = σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T), and hence T satisfies property (gah).

(ii)⇒(i). Suppose that T satisfies property (gah) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T). Then σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Π(T), and hence T satisfies property (VΠ). □

The following example shows that, in general, property (gah) (resp. (ah)) does not imply property (VΠ).

#### Example 2.14

Consider the operator T = 0 defined on the Hilbert space ℓ2(ℕ). Then, σ(T) = $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = {0}, $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = ∅ and Π(T) = {0}. Therefore, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ≠ Π(T) and T does not satisfy property (VΠ). On the other hand, σ(T) ∖ $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = Π(T), that means T satisfies property (gah), in consequence T also satisfies property (ah).

The next result gives the relationship between the properties (VΠ) and (Sb).

#### Corollary 2.15

For TL(X), the following statements are equivalent:

1. T has property (VΠ),

2. T has property (Sab),

3. T has property (Sb).

#### Proof

Follows directly from Theorem 2.13, [3, Theorem 2.4] and [3, Corollary 2.9]. □

The next result gives the relationship between the property (VΠ) and generalized Browder’s theorem.

#### Theorem 2.16

For TL(X), the following statements are equivalent:

1. T has property (VΠ),

2. T satisfies generalized Browder’s theorem and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σBW(T),

3. T satisfies Browder’s theorem and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σBW(T).

#### Proof

(i)⇒(ii). Property (VΠ) implies by Theorem 2.8 that T satisfies property (WΠ), and property (WΠ) implies by [1, Theorem 2.4] that T satisfies generalized Browder’s theorem. Consequently, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) and σ(T) ∖ σBW(T) = Π(T). Therefore, $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σBW(T).

(ii)⇒(i). Assume that T satisfies generalized Browder’s theorem and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σBW(T). Then, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ σBW(T) = Π(T), that means T satisfies property (VΠ).

(ii)⇔(iii). It follows from the equivalence between generalized Browder’s theorem and Browder’s theorem. □

#### Remark 2.17

From Theorem 2.16, property (VΠ) implies generalized Browder’s theorem. However, the converse is not true in general. Consider the operator R in Example 2.9, since R satisfies property (WΠ), then it also satisfies generalized Browder’s theorem, but does not satisfy property (VΠ).

#### Definition 2.18

An operator TL(X) is said to have property (VΠa) if σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T).

#### Example 2.19

Let L be the unilateral left shift operator on ℓ2(ℕ). It is well known that σ(L) = $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(L) = D(0, 1), the closed unit disc onand Πa(L) = ∅. Therefore, σ(L) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(L) = Πa(L), and so L satisfies property (VΠa).

#### Theorem 2.20

Let TL(X). Then T has property (VΠa) if and only if T has property (UWΠa) and σ(T) = σa(T).

#### Proof

Sufficiency: Assume that T satisfies property (VΠa). Then σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ⊆ σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T) and so σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) ⊆ Πa(T).

To show the opposite inclusion Πa(T) ⊆ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T), let λ ∈ Πa(T). Then, λ ∈ Ea(T) and hence λ ∈ σa(T). As T satisfies property (VΠa) and λ ∈ Πa(T), it follows that λ IT is upper semi-Weyl. Therefore, λ ∈ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T). Thus, Πa(T) ⊆ σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) and T satisfies property (UWΠa). Consequently, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T) and σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T). Therefore, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) and σ(T) = σa(T).

Necessity: Suppose that T satisfies property (UWΠa) and σ(T) = σa(T). Then, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T), in consequence T satisfies property (VΠa). □

#### Corollary 2.21

For TL(X), the following statements are equivalent:

1. T has property (VΠa),

2. T has property (VΠ),

3. T has property (Sab),

4. T has property (Sb).

#### Proof

(i)⇒(ii). Suppose that T satisfies property (VΠa). By Theorem 2.20, σ(T) = σa(T), it follows that σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T) = Π(T), hence T satisfies property (VΠ).

(ii)⇒(i). Assume that T satisfies property (VΠ). By Theorem 2.5, σ(T) = σa(T) and so, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) = Πa(T). Therefore, T satisfies property (VΠa).

The rest of the proof follows from Corollary 2.15. □

The next result gives the relationship between property (VΠa) (or equivalently (VΠ)) and property (ZΠa).

#### Theorem 2.22

Let TL(X). Then T has property (VΠa) if and only if T has property (ZΠa) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T).

#### Proof

Sufficiency: Assume that T satisfies property (VΠa). By Corollary 2.21, property (VΠa) is equivalent to property (VΠ), and by Theorem 2.8, property (VΠ) implies that $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T). Consequently, σ(T) ∖ σW(T) = σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Πa(T). Therefore, T satisfies property (ZΠa).

Necessity: Assume that T satisfies property (ZΠa) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T). Then, σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ σW(T) = Πa(T), that means T satisfies property (VΠa). □

Similar to Theorem 2.22, we have the following result.

#### Theorem 2.23

Let TL(X). Then T has property (VΠa) if and only if T has property (gab) and $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σBW(T).

For TL(X), define $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$ (T) = σ(T) ∖ σub(T). The following theorem describes the relationship between a-Browder’s theorem and property (VΠ).

#### Theorem 2.24

For TL(X), the following statements are equivalent:

1. T has property (VΠ),

2. T satisfies a-Browder’s theorem and $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T) = Π(T).

3. T satisfies generalized a-Browder’s theorem and $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T) = Π(T).

#### Proof

(i)⇒(ii) Assume that T satisfies property (VΠ). Then T satisfies property (UWΠ) and Π(T) = $\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$(T) by Theorems 2.5 and 2.10, respectively. Hence σa(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T) = $\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$(T). Consequently, T satisfies a-Browder’s theorem and $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T) = σ(T) ∖ σub(T) = σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = Π(T).

(ii)⇒(i) If T satisfies a-Browder’s theorem and $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T) = Π(T), then σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ σub(T) = $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T) = Π(T). Therefore, T satisfies property (VΠ).

(ii)⇔(iii). It follows from the equivalence between generalized a-Browder’s theorem and a-Browder’s theorem. □

#### Remark 2.25

By Theorem 2.24, property (VE) implies a-Browder’s theorem. However, the converse is not true in general. Indeed, the operator R defined in Example 2.9, does not satisfy property (VΠ), but σa(R) = $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(R) = Γ and $\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$(R) = ∅, it follows that R satisfies a-Browder’s theorem.

Recall that an operator TL(X) is said to have the single valued extension property at λ0 ∈ ℂ (abbreviated SVEP at λ0) if for every open disc 𝔻λ0 ⊆ ℂ centered at λ0, the only analytic function f : 𝔻λ0X which satisfies the equation $(λI−T)f(λ)=0for all λ∈Dλ0$

is f ≡ 0 on 𝔻λ0 (see [12]). The operator T is said to have SVEP, if it has SVEP at every point λ ∈ ℂ. Evidently, every TL(X) has SVEP at each point of the resolvent set ρ (T) := ℂ ∖ σ (T). Moreover, T has SVEP at every point of the boundary ∂σ (T) of the spectrum. In particular, T has SVEP at every isolated point of the spectrum. (See [13] for more details about this concept).

#### Corollary 2.26

If TL(X) has SVEP at each λ ∉ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T), then T has property (VΠ) if and only if Π(T) = $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T).

#### Proof

By [14, Theorem 2.3], the hypothesis T has SVEP at each λ ∉ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) is equivalent to T satisfies a-Browder’s theorem. Therefore, if Π(T) = $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T), then σ(T) ∖ $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σ(T) ∖ σub(T) = $\begin{array}{}{\mathit{\Pi }}_{+}^{0}\end{array}$(T) = Π(T). □

The following three tables summarizes the meaning of various theorems and properties that are related with property (VΠ).

#### Theorem 2.27

Suppose that TL(X) has property (VΠ). Then:

1. $\begin{array}{}{\sigma }_{SB{F}_{+}^{-}}\end{array}$(T) = σBW(T) = $\begin{array}{}{\sigma }_{S{F}_{+}^{-}}\end{array}$(T) = σW(T) = σLD(T) = σD(T) = σub(T) = σb(T) and σ(T) = σa(T).

2. Π0(T) = $\begin{array}{}{\mathit{\Pi }}_{a}^{0}\end{array}$(T) = Π(T) = Πa(T), E0(T) = $\begin{array}{}{E}_{a}^{0}\end{array}$(T) y E(T) = Ea(T).

3. All properties given in Table 1 are equivalent, and T satisfies each of these properties.

4. All properties given in Table 2 are equivalent.

5. All properties given in Table 3 are equivalent.

Table 1

Table 2

Table 3

#### Proof

Since property (VΠ) is equivalent to property (Sab), then (i) and (ii) follows from [3, Theorem 2.31].

(iii) By Theorem 2.5, T satisfies property (UWΠ), and the equivalence between all properties given in Table 1 follows from (i) and (ii).

(iv) and (v) Follows directly from (i) and (ii). □

In the following diagram the arrows signify implications between the Browder type theorems defined above. The numbers near the arrows are references to the results in the present paper (numbers without brackets) or to the bibliography therein (the numbers in square brackets).

$(WΠ)→[1](gB)⟺[6](B)↑2.12↑[8]↑[8](ZΠa)→[4](gab)→[8](ab)↑2.10↑[8]↑[8](WΠ)←[1](UWΠ)←2.5(VΠ)(gb)←[8](b)↑[1]⇕2.15↑[10]↑[10](b)(UWΠa)←2.19(VΠa)⟺2.20(Sb)(gah)⟺[10](ah)↓[19]⇕[1]⇕[3]⇕[10]⇕[10](aB)←[21](SBab)←[21](Bgb)←[3](Sab)→[3](gaz)⟺[9](az)⇕[6]↓[21,17]↓[3](gaB)←[19](gb)(Bb)←[18](Bab)←[4](ZΠa)↓[16]↑[21]⇕[1]↓[18](gB)(Bgb)(WΠ)(ab)$

## Acknowledgement

Research Partially Suported by Vicerrectoría de Investigación, Universidad del Atlántico.

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Accepted: 2017-10-12

Published Online: 2018-04-02

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 289–297, ISSN (Online) 2391-5455,

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