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

#### Definition 2.1

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

#### Example 2.2

*Let T* ∈ *L*(*ℓ*^{2}(ℕ)) *be defined by*
$$\begin{array}{}{\displaystyle T({x}_{1},{x}_{2},{x}_{3},\cdots )=\left(\frac{{x}_{2}}{2},\frac{{x}_{3}}{3},\cdots \right).}\end{array}$$

*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* (*V*_{Π}).

*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 f* ∈ *C*[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* (*V*_{Π}).

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

The next result gives the relationship between the properties (*V*_{E}) and (*V*_{Π}).

#### Theorem 2.3

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{E}),

*T has property* (*V*_{Π}) *and E*(*T*) = *Π*(*T*).

#### Proof

(i)⇒(ii). Suppose that *T* satisfies property (*V*_{E}), 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 (*V*_{Π}) and *E*(*T*) = *Π*(*T*).

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

#### Theorem 2.5

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{Π}),

*T has property* (*UW*_{Π}) *and σ*(*T*) = *σ*_{a}(*T*),

*T has property* (*UW*_{Πa}) *and σ*(*T*) = *σ*_{a}(*T*).

#### Proof

(i)⇒(ii). Suppose that *T* satisfies property (*V*_{Π}) and let λ ∈ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*). Since *σ*_{a}(*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, *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) ⊆ *Π*(*T*).

To show the opposite inclusion *Π*(*T*) ⊆ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*), let λ ∈ *Π*(*T*). Then, λ ∈ *E*_{a}(*T*), it follows that λ ∈ iso œ_{a}(T) and *α*(λ *I* − *T*) > 0, so λ ∈ *σ*_{a}(*T*). As *T* satisfies property (*V*_{Π}) and λ ∈ *Π*(*T*), it follows that λ *I* − *T* is upper semi-Weyl. Therefore, λ ∈ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*). Thus, *Π*(*T*) ⊆ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) and *T* satisfies property (*UW*_{Π}). Consequently, *σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *Π*(*T*) and *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *Π*(*T*). Therefore, *σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) and *σ*(*T*) = *σ*_{a}(*T*).

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

(ii)⇔(iii). Obvious. □

The next example shows that, in general, property (*UW*_{Πa}) does not imply property (*V*_{Π}).

#### Example 2.6

*Let R be the unilateral right shift operator on ℓ*^{2}(ℕ) *and U* ∈ *L*(*ℓ*^{2}(ℕ)) *be defined by*
$$\begin{array}{}U({x}_{1},{x}_{2},{x}_{3},\cdots )=(0,{x}_{2},{x}_{3},\cdots ).\end{array}$$

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

The next example shows that, in general, property (*UW*_{Π}) does not imply property (*V*_{Π}).

#### Example 2.7

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

The next result gives the relationship between the properties (*V*_{Π}) and (*W*_{Π}).

#### Theorem 2.8

*Let T* ∈ *L*(*X*). *Then T has property* (*V*_{Π}) *if and only if T has property* (*W*_{Π}) *and*
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{W}(*T*).

#### Proof

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

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

The next example shows that, in general, property (*W*_{Π}) does not imply property (*V*_{Π}).

#### Example 2.9

*Let R be the unilateral right shift operator on ℓ*^{2}(ℕ). *Since σ*(*R*) = *σ*_{W}(*R*) = **D**(0, 1),
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*R*) = *Γ and* *Π*(*R*) = ∅, *then*
$$\begin{array}{}\sigma (R)\setminus {\sigma}_{W}(R)=\mathit{\Pi}(R),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sigma (R)\setminus {\sigma}_{S{F}_{+}^{-}}(R)\ne \mathit{\Pi}(R).\end{array}$$

*Hence*, *R satisfies property* (*W*_{Π}), *but R does not satisfy property* (*V*_{Π}).

#### Theorem 2.10

*Suppose that T* ∈ *L*(*X*) *has property* (*V*_{Π}). *Then*:

*T has property* (*Z*_{Πa}),

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

#### Proof

Property (*V*_{Π}) implies by Theorem 2.5 that *σ*(*T*) = *σ*_{a}(*T*), and also implies by Theorem 2.8 that
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{W}(*T*). Hence, *σ*(*T*) ∖ *σ*_{W}(*T*) = *σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *Π*(*T*) = *Π*_{a}(*T*) and so *T* satisfies property (*Z*_{Πa}).

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

The next example shows that, in general, property (*Z*_{Πa}) does not imply property (*V*_{Π}).

#### Example 2.11

*Let R be the unilateral right shift operator defined on ℓ*^{2}(ℕ). *Since σ*(*R*) = *σ*_{W}(*R*) = **D**(0, 1), *Π*(*R*) = *Π*_{a}(*R*) = ∅ *and*
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*R*) = *Γ*, *then R satisfies property* (*Z*_{Πa}), *but does not satisfy property* (*V*_{Π}).

#### Theorem 2.13

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{Π}),

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

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

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

*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 (*V*_{Π}). By Theorem 2.5, *T* satisfies property (*UW*_{Πa}). Property (*UW*_{Πa}) 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*) = *Π*_{a} ∖
$\begin{array}{}{\mathit{\Pi}}_{a}^{0}\end{array}$, but by Theorem 2.10, it follow that *Π*_{a} ∖
$\begin{array}{}{\mathit{\Pi}}_{a}^{0}\end{array}$ = ∅ 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 (*V*_{Π}). □

The following example shows that, in general, property (*gah*) (resp. (*ah*)) does not imply property (*V*_{Π}).

#### Example 2.14

*Consider the operator T* = 0 *defined on the Hilbert space ℓ*^{2}(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* (*V*_{Π}). *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 (*V*_{Π}) and (*Sb*).

#### Corollary 2.15

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{Π}),

*T has property* (*Sab*),

*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 (*V*_{Π}) and generalized Browder’s theorem.

#### Theorem 2.16

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{Π}),

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

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

#### Proof

(i)⇒(ii). Property (*V*_{Π}) implies by Theorem 2.8 that *T* satisfies property (*W*_{Π}), and property (*W*_{Π}) 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*) ∖ *σ*_{BW}(*T*) = *Π*(*T*). Therefore,
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{BW}(*T*).

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

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

#### Definition 2.18

*An operator T* ∈ *L*(*X*) *is said to have* property (*V*_{Πa}) *if σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *Π*_{a}(*T*).

#### Example 2.19

*Let L be the unilateral left shift operator on ℓ*^{2}(ℕ). *It is well known that σ*(*L*) =
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*L*) = **D**(0, 1), *the closed unit disc on* ℂ *and Π*_{a}(*L*) = ∅. *Therefore*, *σ*(*L*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*L*) = *Π*_{a}(*L*), *and so L satisfies property* (*V*_{Πa}).

#### Theorem 2.20

*Let T* ∈ *L*(*X*). *Then T has property* (*V*_{Πa}) *if and only if T has property* (*UW*_{Πa}) *and σ*(*T*) = *σ*_{a}(*T*).

#### Proof

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

To show the opposite inclusion *Π*_{a}(*T*) ⊆ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*), let λ ∈ *Π*_{a}(*T*). Then, λ ∈ *E*_{a}(*T*) and hence λ ∈ *σ*_{a}(*T*). As *T* satisfies property (*V*_{Πa}) and λ ∈ *Π*_{a}(*T*), it follows that λ *I* − *T* is upper semi-Weyl. Therefore, λ ∈ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*). Thus, *Π*_{a}(*T*) ⊆ *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) and *T* satisfies property (*UW*_{Πa}). Consequently, *σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *Π*_{a}(*T*) and *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *Π*_{a}(*T*). Therefore, *σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{a}(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) and *σ*(*T*) = *σ*_{a}(*T*).

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

#### Corollary 2.21

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{Πa}),

*T has property* (*V*_{Π}),

*T has property* (*Sab*),

*T has property* (*Sb*).

#### Proof

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

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

The rest of the proof follows from Corollary 2.15. □

The next result gives the relationship between property (*V*_{Πa}) (or equivalently (*V*_{Π})) and property (*Z*_{Πa}).

#### Theorem 2.22

*Let T* ∈ *L*(*X*). *Then T has property* (*V*_{Πa}) *if and only if T has property* (*Z*_{Πa}) *and*
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{W}(*T*).

#### Proof

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

Necessity: Assume that *T* satisfies property (*Z*_{Πa}) and
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{W}(*T*). Then, *σ*(*T*) ∖
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*(*T*) ∖ *σ*_{W}(*T*) = *Π*_{a}(*T*), that means *T* satisfies property (*V*_{Πa}). □

Similar to Theorem 2.22, we have the following result.

#### Theorem 2.23

*Let T* ∈ *L*(*X*). *Then T has property* (*V*_{Πa}) *if and only if T has property* (*gab*) *and*
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{BW}(*T*).

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

#### Theorem 2.24

*For T* ∈ *L*(*X*), *the following statements are equivalent*:

*T has property* (*V*_{Π}),

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

*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 (*V*_{Π}). Then *T* satisfies property (*UW*_{Π}) and *Π*(*T*) =
$\begin{array}{}{\mathit{\Pi}}_{a}^{0}\end{array}$(*T*) by Theorems 2.5 and 2.10, respectively. Hence *σ*_{a}(*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*) ∖ *σ*_{ub}(*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*) ∖ *σ*_{ub}(*T*) =
$\begin{array}{}{\mathit{\Pi}}_{+}^{0}\end{array}$(*T*) = *Π*(*T*). Therefore, *T* satisfies property (*V*_{Π}).

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

#### Corollary 2.26

*If T* ∈ *L*(*X*) *has SVEP at each* λ ∉
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*), *then T has property* (*V*_{Π}) *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*) ∖ *σ*_{ub}(*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 (*V*_{Π}).

#### Theorem 2.27

*Suppose that T* ∈ *L*(*X*) *has property* (*V*_{Π}). *Then*:

$\begin{array}{}{\sigma}_{SB{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{BW}(*T*) =
$\begin{array}{}{\sigma}_{S{F}_{+}^{-}}\end{array}$(*T*) = *σ*_{W}(*T*) = *σ*_{LD}(*T*) = *σ*_{D}(*T*) = *σ*_{ub}(*T*) = *σ*_{b}(*T*) and *σ*(*T*) = *σ*_{a}(*T*).

*Π*^{0}(*T*) =
$\begin{array}{}{\mathit{\Pi}}_{a}^{0}\end{array}$(*T*) = *Π*(*T*) = *Π*_{a}(*T*), *E*^{0}(*T*) =
$\begin{array}{}{E}_{a}^{0}\end{array}$(*T*) *y E*(*T*) = *E*_{a}(*T*).

*All properties given in are equivalent*, *and T satisfies each of these properties*.

*All properties given in are equivalent*.

*All properties given in are equivalent*.

#### Proof

Since property (*V*_{Π}) is equivalent to property (*Sab*), then (i) and (ii) follows from [3, Theorem 2.31].

(iii) By Theorem 2.5, *T* satisfies property (*UW*_{Π}), and the equivalence between all properties given in 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).

$$\begin{array}{}& & & & & & ({W}_{\mathit{\Pi}})& \stackrel{[1]}{\to}& (g\mathcal{B})& {\displaystyle \stackrel{[6]}{\u27fa}}& (\mathcal{B})& & \\ & & & & & & {\uparrow}_{2.12}& & {\uparrow}_{[8]}& & {\uparrow}_{[8]}& & \\ & & & & & & ({Z}_{{\mathit{\Pi}}_{a}})& \stackrel{[4]}{\to}& (gab)& \stackrel{[8]}{\to}& (ab)& & \\ & & & & & & {\uparrow}_{2.10}& & {\uparrow}_{[8]}& & {\uparrow}_{[8]}& & \\ & & ({W}_{\mathit{\Pi}})& \stackrel{[1]}{\leftarrow}& (U{W}_{\mathit{\Pi}})& \stackrel{2.5}{\leftarrow}& ({V}_{\mathit{\Pi}})& & (gb)& \stackrel{[8]}{\leftarrow}& (b)& & \\ & & {\uparrow}_{[1]}& & & & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{2.15}& & {\uparrow}_{[10]}& & {\uparrow}_{[10]}& & \\ (b)& & (U{W}_{{\mathit{\Pi}}_{a}})& \stackrel{2.19}{\leftarrow}& ({V}_{{\mathit{\Pi}}_{a}})& {\displaystyle \stackrel{2.20}{\u27fa}}& (Sb)& & (gah)& {\displaystyle \stackrel{[10]}{\u27fa}}& (ah)& & \\ {\downarrow}^{[19]}& & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{[1]}& & & & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{[3]}& & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{[10]}& & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{[10]}& & \\ (a\mathcal{B})& \stackrel{[21]}{\leftarrow}& (SBab)& \stackrel{[21]}{\leftarrow}& (Bgb)& \stackrel{[3]}{\leftarrow}& (Sab)& \stackrel{[3]}{\to}& (gaz)& {\displaystyle \stackrel{[9]}{\u27fa}}& (az)& & \\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{[6]}& & & & {\downarrow}_{[21,17]}& & {\downarrow}_{[3]}& & & & & & \\ (ga\mathcal{B})& \stackrel{[19]}{\leftarrow}& (gb)& & (Bb)& \stackrel{[18]}{\leftarrow}& (Bab)& \stackrel{[4]}{\leftarrow}& ({Z}_{{\mathit{\Pi}}_{a}})& & & & \\ {\downarrow}_{[16]}& & {\uparrow}_{[21]}& & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\Updownarrow \text{[1]}& & {\downarrow}_{[18]}& & & & & & \\ (g\mathcal{B})& & (Bgb)& & ({W}_{\mathit{\Pi}})& & (ab)& & & & & & \end{array}$$

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