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

Abstract

We shall consider properties which are related to Weyl type theorem for bounded linear operators 𝑇∈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.

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 kernel ker(𝑇), and by 𝛽(𝑇) the codimension of the range ℜ(𝑇). Let SF₊(𝒳)∶={𝑇∈ℒ(𝒳)∶𝛼(𝑇)<∞,ℜ(𝑇)isclosed} (1) be the class of all upper semi-Fredholm operators, and let SF₋(𝒳)∶=𝑇∈ℒ(𝒳)∶𝛽(𝑇)<∞ (2) 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 ind(𝑇)∶=𝛼(𝑇)−𝛽(𝑇). (3) 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 SF⁻₊(𝒳)∶=𝑇∈SF₊(𝒳)∶ind(𝑇)≤0,SF⁺₋(𝒳)∶=𝑇∈SF₋(𝒳)∶ind(𝑇)≥0. (4) The set of Weyl operators is defined by 𝑊(𝒳)∶=SF⁻₊(𝒳)∩SF⁺₋(𝒳)={𝑇∈SF(𝒳)∶ind(𝑇)=0}. (5) The classes of operators defined above generate the following spectra. Denote by 𝜎_𝑎(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆isnotboundedbelow} (6) the approximate point spectrum, and by 𝜎_𝑠(𝑇)∶=𝜆∈ℂ∶𝑇−𝜆isnotsurjective (7) the surjectivity spectrum of 𝑇∈ℒ(𝒳). The Weyl spectrum is defined by 𝜎_𝑤(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆∉𝑊(𝒳)}, (8) the Weyl essential approximate point spectrum is defined by 𝜎_SF⁻₊(𝑇)∶=𝜆∈ℂ∶𝑇−𝜆∉SF⁻₊(𝒳), (9) while the Weyl essential surjectivity spectrum is defined by 𝜎_SF⁺₋(𝑇)∶=𝜆∈ℂ∶𝑇−𝜆∉SF⁺₋(𝒳). (10) Obviously, 𝜎_𝑤(𝑇)=𝜎_SF⁻₊(𝑇)∪𝜎_SF⁺₋(𝑇) and from basic Fredholm theory we have 𝜎_SF⁻₊(𝑇)=𝜎_SF⁺₋𝑇^∗,𝜎_SF⁺₋(𝑇)=𝜎_SF⁻₊𝑇^∗. (11) Note that 𝜎_SF⁻₊(𝑇) is the intersection of all approximate point spectra 𝜎_𝑎(𝑇+𝐾) of compact perturbations 𝐾 of 𝑇, while 𝜎_SF⁺₋(𝑇) is the intersection of all surjectivity spectra 𝜎_𝑠(𝑇+𝐾) of compact perturbations 𝐾 of 𝑇, see, for instance, [1, Theorem 3.65].

Recall that the ascent, 𝑎(𝑇), of an operator 𝑇 is the smallest non-negative integer 𝑝 such that ker(𝑇^𝑝)=ker(𝑇^𝑝+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(𝑇^𝑞)=ℜ(𝑇^𝑞+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 all upper semi-Browder operators is defined by 𝐵₊(𝒳)∶=𝑇∈SF₊(𝒳)∶𝑎(𝑇)<∞, (12) while the class of all lower semi-Browder operators is defined by 𝐵₋(𝒳)∶=𝑇∈SF₊(𝒳)∶𝑑(𝑇)<∞. (13) The class of all Browder operators is defined by 𝐵(𝒳)∶=𝐵₊(𝒳)∩𝐵₋(𝒳)={𝑇∈SF(𝒳)∶𝑎(𝑇),𝑑(𝑇)<∞}. (14) We have 𝐵(𝒳)⊆𝑊(𝒳),𝐵₊(𝒳)⊆SF⁻₊(𝒳),𝐵₋(𝒳)⊆SF⁺₋(𝒳), (15) see [1, Theorem 3.4]. The Browder spectrum of 𝑇∈ℒ(𝒳) is defined by 𝜎_𝑏(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆∉𝐵(𝒳)}, (16) the upper Browder spectrum is defined by 𝜎_𝑢𝑏(𝑇)∶=𝜆∈ℂ∶𝑇−𝜆∉𝐵₊(𝒳), (17) and analogously the lower Browder spectrum is defined by 𝜎_𝑙𝑏(𝑇)∶=𝜆∈ℂ∶𝑇−𝜆∉𝐵₋(𝒳). (18) Clearly, 𝜎_𝑏(𝑇)=𝜎_𝑢𝑏(𝑇)∪𝜎_𝑙𝑏(𝑇) and 𝜎_𝑤(𝑇)⊆𝜎_𝑏(𝑇).

For 𝑇∈𝐵(𝑋) and a nonnegative integer 𝑛 define 𝑇_[𝑛] to be the restriction of 𝑇 to ℜ(𝑇^𝑛) viewed as a map from ℜ(𝑇^𝑛) into ℜ(𝑇^𝑛) (in particular, 𝑇_[0]=𝑇). If for some integer 𝑛 the range space ℜ(𝑇^𝑛) 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 spectrum 𝜎_𝐵𝑊(𝑇) of 𝑇 is defined by 𝜎_𝐵𝑊(𝑇)={𝜆∈ℂ∶𝑇−𝜆isnota𝐵-Weyloperator}.

An operator 𝑇∈ℒ(𝒳) is called Drazin invertible if it has a finite ascent and descent. The Drazin spectrum 𝜎_𝐷(𝑇) of an operator 𝑇 is defined by 𝜎_𝐷(𝑇)={𝜆∈ℂ∶𝑇−𝜆isnotaDrazininvertible}. Define also the set 𝐿𝐷(𝑋) by 𝐿𝐷(𝑋)={𝑇∈ℒ(𝒳)∶𝑎(𝑇)<∞andℜ(𝑇^𝑎(𝑇)+1)isclosed} and 𝜎_𝐿𝐷(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∈𝜎_𝑎(𝑇) is a left pole of 𝑇 if 𝑇−𝜆∈𝐿𝐷(𝑋), and that 𝜆∈𝜎_𝑎(𝑇) is a left pole of 𝑇 of finite rank if 𝜆 is a left pole of 𝑇 and 𝛼(𝑇−𝜆)<∞. Let 𝜋_𝑎(𝑇) denotes the set of all left poles of 𝑇 and let 𝜋⁰_𝑎 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.

Let 𝜋(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)⊂𝜋_𝑎(𝑇) and 𝜋⁰(𝑇)⊂𝜋⁰_𝑎(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 Δ_𝑎(𝑇)=𝜋⁰_𝑎(𝑇). Following [6], we say that generalized Weyl’s theorem holds for 𝑇∈ℒ(𝒳) if Δ^𝑔(𝑇)=𝜎(𝑇)⧵𝜎_𝐵𝑊(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 Δ^𝑔(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.

Let SBF₊(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 𝜎_SBF⁻₊(𝑇)={𝜆∈ℂ∶𝑇−𝜆∉SBF⁻₊(𝑋)}. We say that generalized  𝑎-Weyl’s theorem holds for 𝑇∈ℒ(𝒳) if Δ^𝑔_𝑎(𝑇)=𝜎_𝑎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝐸_𝑎(𝑇), where 𝐸_𝑎(𝑇)={𝜆∈iso𝜎_𝑎(𝑇)∶𝛼(𝑇−𝜆)>0} is the set of all eigenvalues of 𝑇 which are isolated in 𝜎_𝑎(𝑇) and that 𝑇∈ℒ(𝒳) obeys generalized  𝑎-Browder’s theorem if Δ^𝑔_𝑎(𝑇)=𝜋_𝑎(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 𝐸_𝑎(𝑇)=𝜋_𝑎(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 Δ_𝑎(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 Δ^𝑔_𝑎(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 Δ^𝑔_𝑎(𝑇)=𝜋(𝑇), and is said to possess property (𝑏) if Δ_𝑎(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 𝜋⁰_𝑎(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).

The single 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].

Let 𝐻(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 𝑈_𝜆 of 𝜆, the only analytic function 𝑓∶𝑈_𝜆→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]). If 𝑇∈SF_±(𝕏) the following statements are equivalent: (i)𝑇has SVEP at 𝜆₀; (ii)𝑎(𝑇−𝜆₀)<∞; (iii)𝜎_𝑎(𝑇) does not cluster at 𝜆₀; (iv)𝐻₀(𝑇−𝜆₀) is finite dimensional.

By duality we have.

Theorem 2. If 𝑇∈SF_±(𝕏) the following statements are equivalent: (i)𝑇^∗ has SVEP at 𝜆₀; (ii)𝑑(𝑇−𝜆₀)<∞; (iii)𝜎_𝑠(𝑇) does not cluster at 𝜆₀.

In this paper we shall consider properties which are related to Weyl type theorem for bounded linear operators 𝑇∈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. Let 𝑇∈ℒ(𝒳). We say that 𝑇 satisfies (i)property (𝐵) if 𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝜋⁰(𝑇);(ii)property (𝑔𝐵) if 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇).

Theorem 4. Let 𝑇∈ℒ(𝒳). If 𝑇 satisfies property (𝑔𝐵), then 𝑇 satisfies property (𝐵).

Proof. Suppose that 𝑇 satisfies property (𝑔𝐵), then 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇). If 𝜆∈𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇), then 𝜆∈𝜎(𝑇)⧵𝜎_SBF⁻₊(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)⧵𝜎_SF⁻₊(𝑇)⊆𝜋⁰(𝑇). To show the other inclusion, let 𝜆∈𝜋⁰(𝑇) be arbitrary. Then 𝜆∈𝜎(𝑇)⧵𝜎_𝑏(𝑇) and hence 𝜆∈𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇). Therefore, 𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝜋⁰(𝑇), that is, 𝑇 satisfies property (𝐵).

Theorem 5. Let 𝑇∈ℒ(𝒳). Then the following assertions hold. (i)If 𝑇 satisfies property (𝐵), then 𝑇 satisfies property (𝑏).(ii)If 𝑇 satisfies property (𝑔𝐵), then 𝑇 satisfies property (𝑔𝑏).

Proof. (i) Assume that 𝑇 satisfies property (𝐵), then 𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝜋⁰(𝑇). If 𝜆∈Δ_𝑎(𝑇), then 𝜆∈𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝜋⁰(𝑇) and so Δ_𝑎(𝑇)⊆𝜋⁰(𝑇). To show the other inclusion, let 𝜆∈𝜋⁰(𝑇) be an arbitrary. Then 𝜆∈𝜋⁰_𝑎(𝑇)=𝜎_𝑎(𝑇)⧵𝜎_𝑢𝑏(𝑇) and hence 𝜆∈Δ_𝑎(𝑇). Therefore, 𝜋⁰(𝑇)=Δ_𝑎(𝑇), that is, 𝑇 satisfies property (𝑏).
(ii) Assume that 𝑇 satisfies property (𝑔𝐵), then 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇). If 𝜆∈Δ^𝑔_𝑎(𝑇), then 𝜆∈𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇) and so Δ^𝑔_𝑎(𝑇)⊆𝜋(𝑇). To show the other inclusion, let 𝜆∈𝜋(𝑇) be an arbitrary. Then 𝜆∈𝜋_𝑎(𝑇)=𝜎_𝑎(𝑇)⧵𝜎_𝐿𝐷(𝑇) and hence 𝜆∈Δ^𝑔_𝑎(𝑇). Therefore, 𝜋(𝑇)=Δ^𝑔_𝑎(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 operator 𝑇=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, 𝜎_𝑎(𝑇)=𝜎_SF⁻₊(𝑇)=𝜎_SBF⁻₊(𝑇)=𝐶(0,1)∪{0}, where 𝐶(0,1) is the unit circle of ℂ. Since Δ_𝑎(𝑇)=∅=𝜋⁰(𝑇) and Δ^𝑔_𝑎(𝑇)=𝜋(𝑇), then 𝑇 satisfies both property (𝑏) and property (𝑔𝑏). On the other hand, since 𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)≠𝜋⁰(𝑇) and 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)≠𝜋(𝑇), then 𝑇 does not satisfy property (𝐵) nor the property (𝑔𝐵).

As a consequence of Theorem 5, we have.

Corollary 7. Let 𝑇∈ℒ(𝒳). Then the following assertions hold. (i)If 𝑇 satisfies property (𝐵), then 𝑎-Browder’s theorem holds for 𝑇. (ii)If 𝑇 satisfies property (𝑔𝐵), then generalized 𝑎-Browder’s theorem holds for 𝑇.

Theorem 8. Let 𝑇∈ℒ(𝒳). Then the following assertions hold. (i)𝑇 satisfies property (𝐵) if and only if  𝑇 satisfies property (𝑏) and 𝜎(𝑇)=𝜎_𝑎(𝑇).(ii)If 𝑇 satisfies property (𝑔𝐵) if and only if 𝑇 satisfies property (𝑔𝑏) and 𝜎(𝑇)=𝜎_𝑎(𝑇).

Proof. (i) If 𝑇 satisfies property (𝐵), then 𝑇 satisfies property (𝑏) and consequently, Δ_𝑎(𝑇)=𝜋⁰(𝑇) and 𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝜋⁰(𝑇). Hence 𝜎(𝑇)=𝜋⁰(𝑇)∪𝜎_SF⁻₊(𝑇) and 𝜎_𝑎(𝑇)=𝜋⁰(𝑇)∪𝜎_SF⁻₊(𝑇). This implies that 𝜎(𝑇)=𝜎_𝑎(𝑇). Conversely, assume that 𝑇 satisfies property (𝑏) and 𝜎(𝑇)=𝜎_𝑎(𝑇). Then 𝜋⁰(𝑇)=𝜎_𝑎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇). (19) That is, 𝑇 satisfies property (𝐵).
(ii) If 𝑇 satisfies property (𝑔𝐵), then 𝑇 satisfies property (𝑔𝑏) and consequently, Δ^𝑔_𝑎(𝑇)=𝜋(𝑇) and 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇). Hence 𝜎(𝑇)=𝜋(𝑇)∪𝜎_SBF⁻₊(𝑇) and 𝜎_𝑎(𝑇)=𝜋(𝑇)∪𝜎_SBF⁻₊(𝑇). This implies that 𝜎(𝑇)=𝜎_𝑎(𝑇). Conversely, assume that 𝑇 satisfies property (𝑔𝑏) and 𝜎(𝑇)=𝜎_𝑎(𝑇). Then 𝜋(𝑇)=𝜎_𝑎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇). (20) That is, 𝑇 satisfies property (𝑔𝐵).

Theorem 9. Let 𝑇∈ℒ(𝒳). Then the following assertions are equivalent. (i)𝑇 satisfies property (𝑔𝐵). (ii)𝑇 satisfies property (𝐵) and 𝜋(𝑇)=𝜋_𝑎(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)=𝜋_𝑎(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 Δ^𝑔_𝑎(𝑇)=𝜋^𝑎(𝑇). Since 𝑇 satisfies property (𝐵), then Theorem 8 implies that 𝑇 satisfies property (𝑏) and 𝜎(𝑇)=𝜎_𝑎(𝑇). By hypothesis 𝜋(𝑇)=𝜋_𝑎(𝑇), it then follows that 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇) and 𝑇 satisfies property (𝑔𝐵).

Theorem 10. Suppose that 𝑇^∗ has SVEP at every 𝜆∉𝜎_SBF⁻₊(𝑇). Then the following assertions are equivalent: (i)𝐸(𝑇)=𝜋(𝑇); (ii)𝐸_𝑎(𝑇)=𝜋_𝑎(𝑇); (iii)𝐸(𝑇)=𝜋_𝑎(𝑇).
Consequently, property (𝑔𝑤), property (𝑔𝑏), property (𝑔𝐵), generalized 𝑎-Weyl’s theorem and generalized Weyl’s theorem are equivalent for 𝑇.

Proof. Suppose that 𝑇^∗ has SVEP at every 𝜆∉𝜎_SBF⁻₊(𝑇). We prove first the equality 𝜎_SBF⁻₊(𝑇)=𝜎_𝐵𝑊(𝑇). If 𝜆∉𝜎_SBF⁻₊(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∉𝜎_𝐵𝑊(𝑇). Therefore, 𝜎_SBF⁻₊(𝑇)⊆𝜎_𝐵𝑊(𝑇). Since the other inclusion is always verified, we have the equality. Now we prove that 𝜎_𝐷(𝑇)=𝜎_𝐵𝑊(𝑇). Since 𝜎_SBF⁻₊(𝑇)⊆𝜎_SF⁻₊(𝑇) is always verified. Then 𝑇^∗ has SVEP at every 𝜆∉𝜎_SF⁻₊(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 𝜎_𝐵𝑊(𝑇)=𝜎_𝐷(𝑇). On the other hand, as 𝑇^∗ has SVEP at every 𝜆∉𝜎_SBF⁻₊(𝑇), then 𝜎(𝑇)=𝜎_𝑎(𝑇). From this we deduce that 𝐸(𝑇)=𝐸_𝑎(𝑇) and 𝜋_𝑎(𝑇)=𝜎_𝑎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜎(𝑇)⧵𝜎_𝐷(𝑇)=𝜋(𝑇), (21) 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∉𝜎_SBF⁻₊(𝑇) entails that generalized 𝑎-Browder’s theorem (and hence generalized Browder’s theorem) holds for 𝑇, see [20, Corollary 2.7]. Therefor, 𝜋(𝑇)=Δ^𝑔(𝑇)=𝐸(𝑇)=𝜎(𝑇)⧵𝜎_𝐿𝐷(𝑇)=Δ^𝑔_𝑎(𝑇)=𝐸_𝑎(𝑇). (22) 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 that 𝑇 has SVEP at every 𝜆∉𝜎_SBF⁺₋(𝑇). Then the following assertions are equivalent: (i)𝐸(𝑇^∗)=𝜋(𝑇^∗); (ii)𝐸_𝑎(𝑇^∗)=𝜋_𝑎(𝑇^∗); (iii)𝐸(𝑇^∗)=𝜋_𝑎(𝑇^∗).
Consequently, property (𝑔𝐵), property (𝑔𝑤), property (𝑔𝑏), generalized 𝑎-Weyl’s theorem and generalized Weyl’s theorem are equivalent for 𝑇^∗.

Proof. Suppose that 𝑇 has SVEP at every 𝜆∉𝜎_SBF⁺₋(𝑇). We prove first the equality 𝜎_SBF⁺₋(𝑇^∗)=𝜎_𝐵𝑊(𝑇^∗). If 𝜆∉𝜎_SBF⁺₋(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∉𝜎_𝐵𝑊(𝑇). As 𝜎_𝐵𝑊(𝑇)=𝜎_𝐵𝑊(𝑇^∗). Then 𝜆∉𝜎_𝐵𝑊(𝑇^∗). So 𝜎_𝐵𝑊(𝑇^∗)⊆𝜎_SBF⁺₋(𝑇). As 𝜎_SBF⁺₋(𝑇)=𝜎_SBF⁻₊(𝑇^∗), then 𝜎_𝐵𝑊(𝑇^∗)⊆𝜎_SBF⁻₊(𝑇^∗). Since the other inclusion is always verified, it then follows that 𝜎_𝐵𝑊(𝑇^∗)=𝜎_SBF⁻₊(𝑇^∗). Now we show that 𝜎_𝐵𝑊(𝑇^∗)=𝜎_𝐷(𝑇^∗). Since we have always 𝜎_SBF⁺₋(𝑇)⊆𝜎_SF⁺₋(𝑇), then 𝑇 has SVEP at every 𝜆∈𝜎_SF⁺₋(𝑇). Hence 𝑇^∗ satisfies generalized Browder’s theorem. So 𝜎_𝐷(𝑇^∗)=𝜎_𝐵𝑊(𝑇^∗). Finally, we have 𝜎_𝐵𝑊(𝑇^∗)=𝜎_SBF⁻₊(𝑇^∗)=𝜎_𝐷(𝑇^∗) and 𝜎(𝑇^∗)=𝜎_𝑎(𝑇^∗), from which we obtain 𝐸(𝑇^∗)=𝐸_𝑎(𝑇^∗) and 𝜋(𝑇^∗)=𝜋_𝑎(𝑇^∗). The SVEP at every 𝜆∈𝜎_SBF⁺₋(𝑇) ensure by Corollary 2.7 of [20] that generalized 𝑎-Browder’s theorem (and hence generalized Browder’s theorem) holds for 𝑇^∗. Hence 𝜋𝑇^∗=Δ^𝑔𝑇^∗=𝐸𝑇^∗=𝜎𝑇^∗⧵𝜎_𝐿𝐷𝑇^∗=Δ^𝑔_𝑎𝑇^∗=𝐸_𝑎𝑇^∗. (23) 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 that 𝑇^∗ has SVEP at every 𝜆∉𝜎_SF⁻₊(𝑇). Then the following assertions are equivalent: (i)𝐸⁰(𝑇)=𝜋⁰(𝑇); (ii)𝐸⁰_𝑎(𝑇)=𝜋⁰_𝑎(𝑇); (iii)𝐸⁰(𝑇)=𝜋⁰_𝑎(𝑇).
Consequently, property (𝑤), property (𝑏), property (𝐵), property (𝑅),  𝑎-Weyl’s theorem and Weyl’s theorem are equivalent for 𝑇.

Proof. Assume that 𝑇^∗ has SVEP at every 𝜆∉𝜎_SF⁻₊(𝑇). Then it follows from [1, Corollary 2.5] that 𝜎(𝑇)=𝜎_𝑎(𝑇) and by Corollary 3.53 of [1], we then have 𝜎_𝑤(𝑇)=𝜎_𝑏(𝑇)=𝜎_SF⁻₊(𝑇)=𝜎_𝑢𝑏(𝑇). Then the SVEP of 𝑇^∗ at every 𝜆∉𝜎_SF⁻₊(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 𝜋⁰(𝑇)=𝐸⁰(𝑇),𝜋⁰_𝑎(𝑇)=𝐸⁰_𝑎(𝑇),𝐸⁰(𝑇)=𝜋⁰_𝑎(𝑇) (24) are equivalent. Hence 𝜋⁰(𝑇)=Δ(𝑇)=𝐸⁰(𝑇)=𝜎(𝑇)⧵𝜎_𝑢𝑏(𝑇)=Δ_𝑎(𝑇)=𝜋⁰_𝑎(𝑇)=𝐸⁰_𝑎(𝑇). (25) 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 that 𝑇 has SVEP at every 𝜆∉𝜎_SF⁺₋(𝑇). Then the following assertions are equivalent: (i)𝐸⁰(𝑇^∗)=𝜋⁰(𝑇^∗); (ii)𝐸⁰_𝑎(𝑇^∗)=𝜋⁰_𝑎(𝑇^∗); (iii)𝐸⁰(𝑇^∗)=𝜋⁰_𝑎(𝑇^∗).
Consequently, property (𝑤), property (𝑏), property (𝑅), property (𝐵), Weyl’s theorem and 𝑎-Weyl’s theorem are equivalent for 𝑇^∗.

Proof. Assume that 𝑇 has SVEP at every 𝜆∉𝜎_SF⁺₋(𝑇). Then it follows from [1, Corollary 2.5] that 𝜎(𝑇^∗)=𝜎(𝑇)=𝜎_𝑠(𝑇)=𝜎_𝑎(𝑇^∗) and by Corollary 3.53 of [1], we then have 𝜎_𝑤(𝑇^∗)=𝜎_𝑤(𝑇)=𝜎_𝑏(𝑇)=𝜎_SF⁺₋(𝑇)=𝜎_𝑙𝑏(𝑇)=𝜎_𝑢𝑏(𝑇^∗)=𝜎_SF⁻₊(𝑇^∗). Then the SVEP of 𝑇 at every 𝜆∉𝜎_SF⁺₋(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 𝜋⁰𝑇^∗=𝐸⁰𝑇^∗,𝜋⁰_𝑎𝑇^∗=𝐸⁰_𝑎𝑇^∗,𝐸⁰𝑇^∗=𝜋⁰_𝑎𝑇^∗ (26) are equivalent. Hence 𝜋⁰𝑇^∗=Δ𝑇^∗=𝐸⁰𝑇^∗=Δ_𝑎𝑇^∗=𝐸⁰_𝑎𝑇^∗=𝜎_𝑎𝑇^∗⧵𝜎_𝑢𝑏𝑇^∗=𝜋⁰_𝑎𝑇^∗. (27) 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 operator 𝑇∈ℒ(𝒳) is said to satisfy property (𝑎𝑤) if Δ(𝑇)=𝐸⁰_𝑎(𝑇) and is said to satisfy property (𝑔𝑎𝑤) if Δ^𝑔(𝑇)=𝐸_𝑎(𝑇).

Definition 15 (see [24]). A bounded linear operator 𝑇∈ℒ(𝒳) is said to satisfy property (𝑧) if 𝜎(𝑇)⧵𝜎_SF⁻₊(𝑇)=𝐸⁰_𝑎(𝑇) and is said to satisfy property (𝑔𝑧) if 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝐸_𝑎(𝑇).

Theorem 16. Let 𝑇∈ℒ(𝒳). Then 𝑇 satisfies property (𝑔𝑧) if and only if 𝑇 satisfies property (𝑔𝐵) and 𝜋(𝑇)=𝐸_𝑎(𝑇).

Proof. Assume that 𝑇 satisfies property (𝑔𝑧), then 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝐸_𝑎(𝑇). As 𝑇 satisfies property (𝑔𝑧), then we conclude from [24, Theorem 2.7] that 𝑇 satisfies property (𝑔𝑎𝑤),  𝜎_SBF⁻₊(𝑇)=𝜎_𝐵𝑊(𝑇) and from [21, Theorem 3.5] that 𝐸_𝑎(𝑇)=𝜋(𝑇). Therefore, 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇). That is, 𝑇 satisfies property (𝑔𝐵). Conversely, if 𝑇 satisfies property (𝑔𝐵) and 𝜋(𝑇)=𝐸_𝑎(𝑇). Then 𝜎(𝑇)⧵𝜎_SBF⁻₊(𝑇)=𝜋(𝑇)=𝐸_𝑎(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. Let 𝑇∈ℒ(𝒳). Then 𝑇 satisfies property (𝑧) if and only if 𝑇 satisfies property (𝐵) and 𝜋⁰(𝑇)=𝐸⁰_𝑎(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 operator 𝑇∈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 𝜎_𝑎(𝑇) is a pole of the resolvent of 𝑇.

It is easily seen that if 𝑇 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 𝑇ispolaroidifandonlyif𝑇^∗ispolaroid. (28)

From the proof of Theorem 10 we know that if 𝑇^∗ has SVEP, then 𝜎(𝑇)=𝜎_𝑎(𝑇). Therefore, if 𝑇^∗ has SVEP then 𝑇is𝑎-polaroidifandonlyif𝑇ispolaroid. (29) If 𝑇 has SVEP, we know that 𝜎(𝑇^∗)=𝜎_𝑎(𝑇^∗). Therefore, if  𝑇 has SVEP, then 𝑇^∗𝑎-polaroid⟺𝑇^∗polaroid⟺𝑇polaroid. (30)

Theorem 18. Suppose that 𝑇 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 if  𝑇 is 𝑎-polaroid then 𝐸⁰_𝑎(𝑇)=𝜋⁰(𝑇). In fact, if 𝜆∈𝐸⁰_𝑎(𝑇) then 𝜆 is isolated in 𝜎_𝑎(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 𝐸⁰_𝑎(𝑇)⊆𝜋⁰(𝑇). Since the other inclusion is always verified, we have 𝐸⁰_𝑎(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 if  𝑇 is 𝑎-polaroid then 𝐸_𝑎(𝑇)=𝜋(𝑇). In fact, if 𝜆∈𝐸_𝑎(𝑇) then 𝜆 is isolated in 𝜎_𝑎(𝑇) and hence 𝑎(𝑇−𝜆)=𝑑(𝑇−𝜆)<∞. Therefore, 𝜆∈𝜋(𝑇). This shows 𝐸_𝑎(𝑇)⊆𝜋(𝑇). Since we have always 𝜋(𝑇)⊆𝐸_𝑎(𝑇), and so 𝐸_𝑎(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 that 𝑇∈ℒ(𝒳) is polaroid. (i)If  𝑇^∗ has SVEP, then property (𝐵) holds for 𝑇 if and only if  𝑇 satisfies property (𝑧).(ii)If  𝑇 has SVEP, then property (𝐵) holds for 𝑇^∗ if and only if  𝑇 satisfies property (𝑧).

Proof. (i) Since 𝑇^∗ has SVEP, then it follows from [1, Corollary 2.45] that 𝜎(𝑇)=𝜎_𝑎(𝑇), 𝜎_𝑢𝑏(𝑇)=𝜎_SF⁻₊(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 for 𝑇 implies by Corollary 2.45 of [1] that 𝜎(𝑇^∗)=𝜎(𝑇)=𝜎_𝑎(𝑇^∗) and 𝜎_𝑙𝑏(𝑇)=𝜎_𝑢𝑏(𝑇^∗)=𝜎_SF⁺₋(𝑇)=𝜎_SF⁻₊(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 that 𝑇∈ℒ(𝒳) is polaroid. (i)If  𝑇^∗ has SVEP, then property (𝑔𝐵) holds for 𝑇 if and only if  𝑇 satisfies property (𝑔𝑧).(ii)If  𝑇 has SVEP, then property (𝑔𝐵) holds for 𝑇^∗ if and only if  𝑇 satisfies property (𝑔𝑧).

Theorem 21. Suppose that 𝑇 is polaroid and 𝑓∈𝐻(𝜎(𝑇)).(i)If  𝑇^∗ has SVEP, then property (𝐵) holds for 𝑓(𝑇) if and only if property (𝑧) holds for 𝑓(𝑇). (ii)If  𝑇 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 that 𝑇 is polaroid and 𝑓∈𝐻(𝜎(𝑇)).(i)If 𝑇^∗ has SVEP, then property (𝑔𝐵) holds for 𝑓(𝑇) if and only if property (𝑔𝑧) holds for 𝑓(𝑇). (ii)If 𝑇 has SVEP, then property (𝑔𝐵) holds for 𝑓(𝑇^∗) if and only if property (𝑔𝑧) holds for 𝑓(𝑇^∗).

Acknowledgments

The author would like to express their appreciations to the referees for their careful and kind comments.