# Properties(B) and (gB) for Bounded Linear Operators

Authors: M. H. M. Rashid
Journal: Journal of Mathematics
 (2013)

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## Abstract

We shall consider properties which are related to Weyl type theorem for bounded linear operatorsT∈ℒ(𝒳), defined on a complex Banach space 𝒳. These properties, that we call property(B), means that the set of all poles of the resolvent of T of finite rank in the usual spectrum are exactly those points λ of the spectrum for which T-λ is an upper semi-Fredholm with index less than or equal to 0 and we call property(gB), means that the set of all poles of the resolvent of T in the usual spectrum are exactly those points λ of the spectrum for which T-λ is an upper semi-B-Fredholm with index less than or equal to 0. Properties (B) and (gB) are related to a strong variants of classical Weyl’s theorem, the so-called property (b) and property (gb) We shall characterize properties (B) and (gB) in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties (B) and (gB) in the frame of polaroid type operators.


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## Body

## 1. Introduction and Preliminary

Throughout this paper,𝒳 denotes an infinite-dimensional complex Banach space, ℒ(𝒳) the algebra of all bounded linear operators on 𝒳. For an operator T∈ℒ(𝒳) we shall denote by α(T) the dimension of the kernelker(T), and by β(T) the codimension of the rangeℜ(T). Let
(1)SF+(𝒳):={T∈ℒ(𝒳):α(T)<∞,ℜ(T)isclosed}
be the class of all upper semi-Fredholm operators, and let
(2)SF-(𝒳):={T∈ℒ(𝒳):β(T)<∞}
be the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by SF±(𝒳):=SF+(𝒳)∪SF-(𝒳), while the class of all Fredholm operators is defined by SF(𝒳):=SF+(𝒳)∩SF-(𝒳). If T∈SF±(𝒳), the index of T is defined by
(3)ind(T):=α(T)-β(T).
Recall that a bounded operator T is said bounded below if it injective and has closed range. Evidently, if T is bounded below then T∈SF+(𝒳) and ind(T)≤0. Define
(4)SF+-(𝒳):={T∈SF+(𝒳):ind(T)≤0},SF-+(𝒳):={T∈SF-(𝒳):ind(T)≥0}.
The set of Weyl operators is defined by
(5)W(𝒳):=SF+-(𝒳)∩SF-+(𝒳)={T∈SF(𝒳):ind(T)=0}.
The classes of operators defined above generate the following spectra. Denote by
(6)σa(T):={λ∈ℂ:T-λisnotboundedbelow}
the approximate point spectrum, and by
(7)σs(T):={λ∈ℂ:T-λisnotsurjective}
the surjectivity spectrum of T∈ℒ(𝒳). The Weyl spectrum is defined by
(8)σw(T):={λ∈ℂ:T-λ∉W(𝒳)},
the Weyl essential approximate point spectrum is defined by
(9)σSF+-(T):={λ∈ℂ:T-λ∉SF+-(𝒳)},
while the Weyl essential surjectivity spectrum is defined by
(10)σSF-+(T):={λ∈ℂ:T-λ∉SF-+(𝒳)}.
Obviously, σw(T)=σSF+-(T)∪σSF-+(T) and from basic Fredholm theory we have
(11)σSF+-(T)=σSF-+(T*),σSF-+(T)=σSF+-(T*).
Note that σSF+-(T) is the intersection of all approximate point spectra σa(T+K) of compact perturbations K of T, while σSF-+(T) is the intersection of all surjectivity spectra σs(T+K) of compact perturbations K of T, see, for instance, [1, Theorem 3.65].Recall that theascent, a(T), of an operator T is the smallest non-negative integer p such that ker(Tp)=ker(Tp+1). If such integer does not exist we put a(T)=∞. Analogously, the descent, d(T), of an operator T is the smallest non-negative integer q such that ℜ(Tq)=ℜ(Tq+1), and if such integer does not exist we put d(T)=∞. It is well known that if a(T) and d(T) are both finite then a(T)=d(T) [2, Proposition 1.49]. Moreover, 0<a(T-λ)=d(T-λ)<∞ precisely when λ is a pole of the resolvent of T, see Dowson [2, Theorem 1.54].The class of allupper semi-Browder operators is defined by
(12)B+(𝒳):={T∈SF+(𝒳):a(T)<∞},
while the class of all lower semi-Browder operators is defined by
(13)B-(𝒳):={T∈SF+(𝒳):d(T)<∞}.
The class of all Browder operators is defined by
(14)B(𝒳)≔B+(𝒳)∩B-(𝒳)={T∈SF(𝒳):a(T),d(T)<∞}.
We have
(15)B(𝒳)⊆W(𝒳),B+(𝒳)⊆SF+-(𝒳),B-(𝒳)⊆SF-+(𝒳),
see [1, Theorem 3.4]. The Browder spectrum of T∈ℒ(𝒳) is defined by
(16)σb(T):={λ∈ℂ:T-λ∉B(𝒳)},
the upper Browder spectrum is defined by
(17)σub(T):={λ∈ℂ:T-λ∉B+(𝒳)},
and analogously the lower Browder spectrum is defined by
(18)σlb(T):={λ∈ℂ:T-λ∉B-(𝒳)}.
Clearly, σb(T)=σub(T)∪σlb(T) and σw(T)⊆σb(T).ForT∈B(X) and a nonnegative integer n define T[n] to be the restriction of T to ℜ(Tn) viewed as a map from ℜ(Tn) into ℜ(Tn) (in particular, T[0]=T). If for some integer n the range space ℜ(Tn) is closed and T[n] is an upper (a lower) semi-Fredholm operator, then T is called an upper (a lower) semi-B-Fredholm operator. In this case the index of T is defined as the index of the semi-Fredholm operator T[n], see [3]. Moreover, if T[n] is a Fredholm operator, then T is called a  B-Fredholm operator. A semi-B-Fredholm operator is an upper or a lower semi-B-Fredholm operator. An operator T is said to be a  B-Weyl operator if it is a  B-Fredholm operator of index zero. The  B-Weyl spectrumσBW(T) of T is defined by σBW(T)={λ∈ℂ:T-λisnotaB-Weyloperator}.An operatorT∈ℒ(𝒳) is called Drazin invertible if it has a finite ascent and descent. The Drazin spectrumσD(T) of an operator T is defined by σD(T)={λ∈ℂ:T-λisnotaDrazininvertible}. Define also the set LD(X) by LD(X)={T∈ℒ(𝒳):a(T)<∞andℜ(Ta(T)+1)isclosed} and σLD(T)={λ∈ℂ:T-λ∉LD(X)}. Following [4], an operator T∈ℒ(𝒳) is said to be left Drazin invertible if T∈LD(X). We say that λ∈σa(T) is a left pole of T if T-λ∈LD(X), and that λ∈σa(T) is a left pole of T of finite rank if λ is a left pole of T and α(T-λ)<∞. Let πa(T) denotes the set of all left poles of T and let πa0 denotes the set of all left poles of T of finite rank. From Theorem 2.8 of [4] it follows that if T∈ℒ(𝒳) is left Drazin invertible, then T is an upper semi-B-Fredholm operator of index less than or equal to 0.Letπ(T) be the set of all poles of the resolvent of T and let π0(T) be the set of all poles of the resolvent of T of finite rank, that is π0(T)={λ∈π(T):α(T-λ)<∞}. According to [5], a complex number λ is a pole of the resolvent of T if and only if 0<max{a(T-λ),d(T-λ)}<∞. Moreover, if this is true then a(T-λ)=d(T-λ).According also to [5], the space ℜ((T-λ)a(T-λ)+1) is closed for each λ∈π(T). Hence we have always π(T)⊂πa(T) and π0(T)⊂πa0(T). We say that Browder’s theorem holds for T∈ℒ(𝒳) if Δ(T)=π0(T), and that  a-Browder’s theorem holds for T∈ℒ(𝒳) if Δa(T)=πa0(T). Following [6], we say that generalized Weyl’s theorem holds for T∈ℒ(𝒳) if Δg(T)=σ(T)∖σBW(T)=E(T), where E(T)={λ∈isoσ(T):α(T-λ)>0} is the set of all isolated eigenvalues of T, and that generalized Browder’s theorem holds for T∈ℒ(𝒳) if Δg(T)=π(T). It is proved in Theorem 2.1 of [7] that generalized Browder’s theorem is equivalent to Browder’s theorem. In [4, Theorem 3.9], it is shown that an operator satisfying generalized Weyl’s theorem satisfies also Weyl’s theorem, but the converse does not hold in general. Nonetheless and under the assumption E(T)=π(T), it is proved in Theorem 2.9 of   [8] that generalized Weyl’s theorem is equivalent to Weyl’s theorem.LetSBF+(X) be the class of all upper semi-B-Fredholm operators, SBF+-(X)={T∈SBF+(X):ind(T)≤0}. The upper  B-Weyl spectrum of T is defined by σSBF+-(T)={λ∈ℂ:T-λ∉SBF+-(X)}. We say that generalized  a-Weyl’s theorem holds for T∈ℒ(𝒳) if Δag(T)=σa(T)∖σSBF+-(T)=Ea(T), where Ea(T)={λ∈isoσa(T):α(T-λ)>0} is the set of all eigenvalues of T which are isolated in σa(T) and that T∈ℒ(𝒳) obeys generalized  a-Browder’s theorem if Δag(T)=πa(T). It is proved in [7, Theorem 2.2] that generalized  a-Browder’s theorem is equivalent to  a-Browder’s theorem, and it is known from [4, Theorem 3.11] that an operator satisfying generalized  a-Weyl’s theorem satisfies  a-Weyl’s theorem, but the converse does not hold in general and under the assumption Ea(T)=πa(T) it is proved in [8, Theorem 2.10] that generalized  a-Weyl’s theorem is equivalent to  a-Weyl’s theorem.Following [9], we say that T∈ℒ(𝒳) possesses property(w) if Δa(T)=E0(T). The property (w) has been studied in [1]. In Theorem 2.8 of [10], it is shown that property (w) implies Weyl’s theorem, but the converse is not true in general. We say that T∈ℒ(𝒳) possesses property(gw) if Δag(T)=E(T). Property (gw) has been introduced and studied in [11, 12]. Property (gw) extends property (w) to the context of  B-Fredholm theory, and it is proved in [11] that an operator possessing property (gw) possesses property (w) but the converse is not true in general. According to [13], an operator T∈ℒ(𝒳) is said to possess property(gb) if Δag(T)=π(T), and is said to possess property(b) if Δa(T)=π0(T). It is shown in Theorem 2.3 of [13] that an operator possessing property (gb) possesses property (b) but the converse is not true in general, see also [14]. Following [15], we say an operator T∈ℒ(𝒳) is said to be satisfies property(R) if πa0(T)=E0(T). In Theorem 2.4 of [15], it is shown that T satisfies property (w) if and only if T satisfies a-Browder’s theorem and T satisfies property (R).Thesingle valued extension property plays an important role in local spectral theory, see the recent monograph of Laursen and Neumann [16] and Aiena [1]. In this article we shall consider the following local version of this property, which has been studied in recent papers, [10, 17] and previously by Finch [18].LetH(σ(T)) be the space of all functions that analytic in an open neighborhoods of σ(T). Following [18] we say that T∈ℒ(𝒳) has the single-valued extension property (SVEP) at point λ∈ℂ if for every open neighborhood Uλ of λ, the only analytic function f:Uλ→𝒳 which satisfies the equation (T-μ)f(μ)=0 is the constant function f≡0. It is well-known that T∈ℒ(𝒳) has SVEP at every point of the resolvent ρ(T):=ℂ∖σ(T). Moreover, from the identity Theorem for analytic function it easily follows that T∈ℒ(𝒳) has SVEP at every point of the boundary ∂σ(T) of the spectrum. In particular, T has SVEP at every isolated point of σ(T). In [17, Proposition 1.8], Laursen proved that if T is of finite ascent, then T has SVEP.Theorem 1 (see [19, Theorem 1.3]).
IfT∈SF±(𝕏) the following statements are equivalent: (i)
Thas SVEP at λ0;(ii)
a(T-λ0)<∞;(iii)
σa(T) does not cluster at λ0;(iv)
H0(T-λ0) is finite dimensional.By duality we have.Theorem 2.
IfT∈SF±(𝕏) the following statements are equivalent: (i)
T* has SVEP at λ0;(ii)
d(T-λ0)<∞;(iii)
σs(T) does not cluster at λ0.In this paper we shall consider properties which are related to Weyl type theorem for bounded linear operatorsT∈ℒ(𝒳), defined on a complex Banach space 𝒳. These properties, that we call property(B), means that the set of all poles of the resolvent of T of finite rank in the usual spectrum are exactly those points λ of the spectrum for which T-λ is an upper semi-Fredholm with index less than or equal to 0 (see Definition 3) and we call property(gB), means that the set of all poles of the resolvent of T in the usual spectrum are exactly those points λ of the spectrum for which T-λ is an upper semi-B-Fredholm with index less than or equal to 0 (see Definition 3). Properties (B) and (gB) are related to a strong variants of classical Weyl’s theorem, the so-called property (b) and property (gb) introduced by Berkani and Zariouh [13] and more extensively studied in recent papers [12, 14, 20, 21]. We shall characterize properties (B) and (gB) in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties (B) and (gB) in the frame of polaroid type operators.


## 2. Properties (B) and (gB)

Definition 3.
LetT∈ℒ(𝒳). We say that T satisfies (i)
property(B) if σ(T)∖σSF+-(T)=π0(T);(ii)
property(gB) if σ(T)∖σSBF+-(T)=π(T).Theorem 4.
LetT∈ℒ(𝒳). If T satisfies property (gB), then T satisfies property (B).Proof.
Suppose thatT satisfies property (gB), then σ(T)∖σSBF+-(T)=π(T). If λ∈σ(T)∖σSF+-(T), then λ∈σ(T)∖σSBF+-(T)=π(T). Since T-λ is an upper semi-Fredholm, then α(T-λ)<∞. As λ∈π(T), then a(T-λ)=d(T-λ)<∞. So, it follows from [1, Theorem 3.4] that β(T-λ)=α(T-λ)<∞. Hence λ∈π0(T) and this implies that σ(T)∖σSF+-(T)⊆π0(T). To show the other inclusion, let λ∈π0(T) be arbitrary. Then λ∈σ(T)∖σb(T) and hence λ∈σ(T)∖σSF+-(T). Therefore, σ(T)∖σSF+-(T)=π0(T), that is, T satisfies property (B).Theorem 5.
LetT∈ℒ(𝒳). Then the following assertions hold. (i)
IfT satisfies property (B), then T satisfies property (b).(ii)
IfT satisfies property (gB), then T satisfies property (gb).Proof.
(i) Assume thatT satisfies property (B), then σ(T)∖σSF+-(T)=π0(T). If λ∈Δa(T), then λ∈σ(T)∖σSF+-(T)=π0(T) and so Δa(T)⊆π0(T). To show the other inclusion, let λ∈π0(T) be an arbitrary. Then λ∈πa0(T)=σa(T)∖σub(T) and hence λ∈Δa(T). Therefore, π0(T)=Δa(T), that is, T satisfies property (b).
(ii) Assume thatT satisfies property (gB), then σ(T)∖σSBF+-(T)=π(T). If λ∈Δag(T), then λ∈σ(T)∖σSBF+-(T)=π(T) and so Δag(T)⊆π(T). To show the other inclusion, let λ∈π(T) be an arbitrary. Then λ∈πa(T)=σa(T)∖σLD(T) and hence λ∈Δag(T). Therefore, π(T)=Δag(T), that is, T satisfies property (gb).The following example shows the converse of the previous theorem does not hold in general.Example 6.
Consider the operatorT=R⊕S that defined on 𝒳=ℓ2(ℕ)⊕ℓ2(ℕ), where R is the right unilateral shift operator and S(x1,x2,…)=(x2/2,x3/3,…). Then σ(T)=D(0,1), where D(0,1) is the unit disc of ℂ. Hence, isoσ(T)=∅ and so, E0(T)=π0(T)=π(T)=E(T)=∅. Moreover, σa(T)=σSF+-(T)=σSBF+-(T)=C(0,1)∪{0}, where C(0,1) is the unit circle of ℂ. Since Δa(T)=∅=π0(T) and Δag(T)=π(T), then T satisfies both property (b) and property (gb). On the other hand, since σ(T)∖σSF+-(T)≠π0(T) and σ(T)∖σSBF+-(T)≠π(T), then T does not satisfy property (B) nor the property (gB).As a consequence of Theorem5, we have.Corollary 7.
LetT∈ℒ(𝒳). Then the following assertions hold. (i)
IfT satisfies property (B), then a-Browder’s theorem holds for T.(ii)
IfT satisfies property (gB), then generalized a-Browder’s theorem holds for T.Theorem 8.
LetT∈ℒ(𝒳). Then the following assertions hold. (i)
T satisfies property (B) if and only if  T satisfies property (b) and σ(T)=σa(T).(ii)
IfT satisfies property (gB) if and only if T satisfies property (gb) and σ(T)=σa(T).Proof.
(i) IfT satisfies property (B), then T satisfies property (b) and consequently, Δa(T)=π0(T) and σ(T)∖σSF+-(T)=π0(T). Hence σ(T)=π0(T)∪σSF+-(T) and σa(T)=π0(T)∪σSF+-(T). This implies that σ(T)=σa(T). Conversely, assume that T satisfies property (b) and σ(T)=σa(T). Then
(19)π0(T)=σa(T)∖σSF+-(T)=σ(T)∖σSF+-(T).
That is, T satisfies property (B).
(ii) IfT satisfies property (gB), then T satisfies property (gb) and consequently, Δag(T)=π(T) and σ(T)∖σSBF+-(T)=π(T). Hence σ(T)=π(T)∪σSBF+-(T) and σa(T)=π(T)∪σSBF+-(T). This implies that σ(T)=σa(T). Conversely, assume that T satisfies property (gb) and σ(T)=σa(T). Then
(20)π(T)=σa(T)∖σSBF+-(T)=σ(T)∖σSBF+-(T).
That is, T satisfies property (gB).Theorem 9.
LetT∈ℒ(𝒳). Then the following assertions are equivalent. (i)
T satisfies property (gB).(ii)
T satisfies property (B) and π(T)=πa(T).Proof.
(i)⇒(ii) Assume that T satisfies property (gB), then T satisfies property (B). As T satisfies property (gB), then T satisfies property (gb), and hence it follows from [13, Corollary 2.9] that T satisfies generalized a-Browder’s theorem and π(T)=πa(T).
(ii)⇒(i) Assume that T satisfies property (B) and then by Corollary 7, T satisfies a-Browder’s theorem. As we know from [7, Theorem 2.2] that a-Browder’s theorem is equivalent to generalized a-Browder’s theorem, then T satisfies generalized a-Browder’s theorem. Hence we have Δag(T)=πa(T). Since T satisfies property (B), then Theorem 8 implies that T satisfies property (b) and σ(T)=σa(T). By hypothesis π(T)=πa(T), it then follows that σ(T)∖σSBF+-(T)=π(T) and T satisfies property (gB).Theorem 10.
Suppose thatT* has SVEP at every λ∉σSBF+-(T). Then the following assertions are equivalent: (i)
E(T)=π(T);(ii)
Ea(T)=πa(T);(iii)
E(T)=πa(T).
Consequently, property(gw), property (gb), property (gB), generalized a-Weyl’s theorem and generalized Weyl’s theorem are equivalent for T.Proof.
Suppose thatT* has SVEP at every λ∉σSBF+-(T). We prove first the equality σSBF+-(T)=σBW(T). If λ∉σSBF+-(T) then T-λ is an upper semi-B-Fredholm operator and ind(T-λ)≤0. As T* has SVEP, then it follows from Corollary 2.8 of [22] that T-λ is a  B-Weyl operator and so λ∉σBW(T). Therefore, σSBF+-(T)⊆σBW(T). Since the other inclusion is always verified, we have the equality. Now we prove that σD(T)=σBW(T). Since σSBF+-(T)⊆σSF+-(T) is always verified. Then T* has SVEP at every λ∉σSF+-(T). This implies that T satisfies Browder’s theorem. As we know from Theorem 2.1 of [7] that Browder’s theorem is equivalent to generalized Browder’s theorem, we have σBW(T)=σD(T). On the other hand, as T* has SVEP at every λ∉σSBF+-(T), then σ(T)=σa(T). From this we deduce that E(T)=Ea(T) and
(21)πa(T)=σa(T)∖σSBF+-(T)=σ(T)∖σD(T)=π(T),
from which the equivalence of (i), (ii) and (iii) easily follows. To show the last statement observed that the SVEP of T* at the points λ∉σSBF+-(T) entails that generalized a-Browder’s theorem (and hence generalized Browder’s theorem) holds for T, see [20, Corollary 2.7]. Therefor,
(22)π(T)=Δg(T)=E(T)=σ(T)∖σLD(T)=Δag(T)=Ea(T).
That is, property (gB), property (gb), property (gw), generalized a-Weyl’s theorem and generalized Weyl’s theorem are equivalent for T.Dually, we have.Theorem 11.
Suppose thatT has SVEP at every λ∉σSBF-+(T). Then the following assertions are equivalent: (i)
E(T*)=π(T*);(ii)
Ea(T*)=πa(T*);(iii)
E(T*)=πa(T*).
Consequently, property(gB), property (gw), property (gb), generalized a-Weyl’s theorem and generalized Weyl’s theorem are equivalent for T*.Proof.
Suppose thatT has SVEP at every λ∉σSBF-+(T). We prove first the equality σSBF-+(T*)=σBW(T*). If λ∉σSBF-+(T) then T-λ is a lower semi-B-Fredholm operator and ind(T-λ)≥0. As T has SVEP, then it follows from Theorem 2.5 of [22] that T-λ is a  B-Weyl operator and so λ∉σBW(T). As σBW(T)=σBW(T*). Then λ∉σBW(T*). So σBW(T*)⊆σSBF-+(T). As σSBF-+(T)=σSBF+-(T*), then σBW(T*)⊆σSBF+-(T*). Since the other inclusion is always verified, it then follows that σBW(T*)=σSBF+-(T*). Now we show that σBW(T*)=σD(T*). Since we have always σSBF-+(T)⊆σSF-+(T), then T has SVEP at every λ∈σSF-+(T). Hence T* satisfies generalized Browder’s theorem. So σD(T*)=σBW(T*). Finally, we have σBW(T*)=σSBF+-(T*)=σD(T*) and σ(T*)=σa(T*), from which we obtain E(T*)=Ea(T*) and π(T*)=πa(T*). The SVEP at every λ∈σSBF-+(T) ensure by Corollary 2.7 of [20] that generalized a-Browder’s theorem (and hence generalized Browder’s theorem) holds for T*. Hence
(23)π(T*)=Δg(T*)=E(T*)=σ(T*)∖σLD(T*)=Δag(T*)=Ea(T*).
That is, property (gb), property (gB), property (gw), generalized a-Weyl’s theorem and generalized Weyl’s theorem are equivalent for T*.Corollary 12.
Suppose thatT* has SVEP at every λ∉σSF+-(T). Then the following assertions are equivalent: (i)
E0(T)=π0(T);(ii)
Ea0(T)=πa0(T);(iii)
E0(T)=πa0(T).
Consequently, property(w), property (b), property (B), property (R),  a-Weyl’s theorem and Weyl’s theorem are equivalent for T.Proof.
Assume thatT* has SVEP at every λ∉σSF+-(T). Then it follows from [1, Corollary 2.5] that σ(T)=σa(T) and by Corollary 3.53 of [1], we then have σw(T)=σb(T)=σSF+-(T)=σub(T). Then the SVEP of T* at every λ∉σSF+-(T) entails that a-Browder’s theorem (and hence Browder’s theorem) holds for T, see [23, Theorem 2.3]. Then it follows by Theorem 2.19 of [15] that
(24)π0(T)=E0(T),πa0(T)=Ea0(T),E0(T)=πa0(T)
are equivalent. Hence
(25)π0(T)=Δ(T)=E0(T)=σ(T)∖σub(T)=Δa(T)=πa0(T)=Ea0(T).
Therefore, property (w), property (b), property (B), property (R),  a-Weyl’s theorem and Weyl’s theorem are equivalent for T.Dually, we have.Corollary 13.
Suppose thatT has SVEP at every λ∉σSF-+(T). Then the following assertions are equivalent: (i)
E0(T*)=π0(T*);(ii)
Ea0(T*)=πa0(T*);(iii)
E0(T*)=πa0(T*).
Consequently, property(w), property (b), property (R), property (B), Weyl’s theorem and a-Weyl’s theorem are equivalent for T*.Proof.
Assume thatT has SVEP at every λ∉σSF-+(T). Then it follows from [1, Corollary 2.5] that σ(T*)=σ(T)=σs(T)=σa(T*) and by Corollary 3.53 of [1], we then have σw(T*)=σw(T)=σb(T)=σSF-+(T)=σlb(T)=σub(T*)=σSF+-(T*). Then the SVEP of T at every λ∉σSF-+(T) entails that a-Browder’s theorem (and hence Browder’s theorem) holds for T*, see [23, Theorem 2.3]. Then it follows by Theorem 2.20 of [15] that
(26)π0(T*)=E0(T*),πa0(T*)=Ea0(T*),E0(T*)=πa0(T*)
are equivalent. Hence
(27)π0(T*)=Δ(T*)=E0(T*)=Δa(T*)=Ea0(T*)=σa(T*)∖σub(T*)=πa0(T*).
Therefore, property (w), property (b), property (R), property (B), Weyl’s theorem and a-Weyl’s theorem are equivalent for T*.Definition 14 (see [21]).
A bounded linear operatorT∈ℒ(𝒳) is said to satisfy property (aw) if Δ(T)=Ea0(T) and is said to satisfy property (gaw) if Δg(T)=Ea(T).Definition 15 (see [24]).
A bounded linear operatorT∈ℒ(𝒳) is said to satisfy property (z) if σ(T)∖σSF+-(T)=Ea0(T) and is said to satisfy property (gz) if σ(T)∖σSBF+-(T)=Ea(T).Theorem 16.
LetT∈ℒ(𝒳). Then T satisfies property (gz) if and only if T satisfies property (gB) and π(T)=Ea(T).Proof.
Assume thatT satisfies property (gz), then σ(T)∖σSBF+-(T)=Ea(T). As T satisfies property (gz), then we conclude from [24, Theorem 2.7] that T satisfies property (gaw),  σSBF+-(T)=σBW(T) and from [21, Theorem 3.5] that Ea(T)=π(T). Therefore, σ(T)∖σSBF+-(T)=π(T). That is, T satisfies property (gB). Conversely, if T satisfies property (gB) and π(T)=Ea(T). Then σ(T)∖σSBF+-(T)=π(T)=Ea(T). That is, T satisfies property (gz).Similarly we have the following result in the case of property(B), which we give without proof.Theorem 17.
LetT∈ℒ(𝒳). Then T satisfies property (z) if and only if T satisfies property (B) and π0(T)=Ea0(T).


## 3. Properties (B) and (gB) for Polaroid Type Operators

In this section we consider classes of operators for which the isolated points of the spectrum are poles of the resolvent.An operatorT∈ℒ(𝒳) is said to be polaroid if isolated point of σ(T) is a pole of the resolvent of T. T∈ℒ(𝒳) is said to be a-polaroid if every isolated of σa(T) is a pole of the resolvent of T.It is easily seen that ifT is a-polaroid, then T is polaroid, while in general the converse is not true. It is well known that λ is a pole of the resolvent of T if and only if λ is a pole of the resolvent of T*. Since σ(T)=σ(T*) we then have
(28)TispolaroidifandonlyifT*ispolaroid.From the proof of Theorem10 we know that if T* has SVEP, then σ(T)=σa(T). Therefore, if T* has SVEP then
(29)Tisa-polaroidifandonlyifTispolaroid.
If T has SVEP, we know that σ(T*)=σa(T*). Therefore, if  T has SVEP, then
(30)T*a-polaroid⟺T*polaroid⟺Tpolaroid.Theorem 18.
Suppose thatT is a-polaroid. Then the following assertions hold. (i)
T satisfies property (z) if and only if T satisfies property (B).(ii)
T satisfies property (gz) if and only if  T satisfies property (gB).Proof.
(i) Note first that ifT is a-polaroid then Ea0(T)=π0(T). In fact, if λ∈Ea0(T) then λ is isolated in σa(T) and hence a(T-λ)=d(T-λ)<∞. Moreover, α(T-λ)<∞, so by Theorem 3.4 of [1] it follows that β(T-λ) is also finite, thus λ∈π0(T). This shows Ea0(T)⊆π0(T). Since the other inclusion is always verified, we have Ea0(T)=π0(T). Therefore, it follows from Theorem 17 that T satisfies property (z) if and only if T satisfies property (B).
(ii) Note first that ifT is a-polaroid then Ea(T)=π(T). In fact, if λ∈Ea(T) then λ is isolated in σa(T) and hence a(T-λ)=d(T-λ)<∞. Therefore, λ∈π(T). This shows Ea(T)⊆π(T). Since we have always π(T)⊆Ea(T), and so Ea(T)=π(T). Therefore, it follows from Theorem 16 that T satisfies property (gz) if and only if T satisfies property (gB).Theorem 19.
Suppose thatT∈ℒ(𝒳) is polaroid. (i)
IfT* has SVEP, then property (B) holds for T if and only if  T satisfies property (z).(ii)
IfT has SVEP, then property (B) holds for T* if and only if  T satisfies property (z).Proof.
(i) SinceT* has SVEP, then it follows from [1, Corollary 2.45] that σ(T)=σa(T), σub(T)=σSF+-(T), see also [25, Theorem 1.5]. Since T is polaroid and T* has SVEP, then by equivalence (29), we have T is a-polaroid and so the result follows now from Theorem 18.
(ii) The SVEP forT implies by Corollary 2.45 of [1] that σ(T*)=σ(T)=σa(T*) and σlb(T)=σub(T*)=σSF-+(T)=σSF+-(T*). Since T* is polaroid by equivalence (28) and T has SVEP, then T* is a-polaroid and so the result follows now from Theorem 18.Similarly we have the following result in the case of property(gB), which we give without proof.Theorem 20.
Suppose thatT∈ℒ(𝒳) is polaroid. (i)
IfT* has SVEP, then property (gB) holds for T if and only if  T satisfies property (gz).(ii)
IfT has SVEP, then property (gB) holds for T* if and only if  T satisfies property (gz).Theorem 21.
Suppose thatT is polaroid and f∈H(σ(T)).(i)
IfT* has SVEP, then property (B) holds for f(T) if and only if property (z) holds for f(T).(ii)
IfT has SVEP, then property (B) holds for f(T*) if and only if property (z) holds for f(T*).Proof.
(i) It follows from Lemma 3.11 of [26] that f(T) is polaroid. By Theorem 2.40 of [1], we have f(T*) has SVEP, hence from equivalence (29) we conclude that f(T) is a-polaroid and hence it then follows by Theorem 19 that property (B) holds for f(T) if and only if property (z) holds for f(T).
(ii) From the equivalence (28) we know that T* is polaroid and hence by Lemma 3.11 of [26] that f(T*) is polaroid. By Theorem 2.40 of [1], we have f(T) has SVEP, hence from equivalence (29) we conclude that f(T*) is a-polaroid and hence it then follows by Theorem 19 that property (B) holds for f(T*) if and only if property (z) holds for f(T*).Similarly we have the following result in the case of property(gB), which we give without proof.Theorem 22.
Suppose thatT is polaroid and f∈H(σ(T)).(i)
IfT* has SVEP, then property (gB) holds for f(T) if and only if property (gz) holds for f(T).(ii)
IfT has SVEP, then property (gB) holds for f(T*) if and only if property (gz) holds for f(T*).



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