Study of the property (bz) using local spectral theory methods

Abstract For a bounded linear operator, by local spectral theory methods, we study the property (bz), which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of X, also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed.


Introduction
It is well known that the spectral theory of linear operators (in special cases) has numerous applications in different fields; among others, we can mention its use in artificial intelligence, which develops the analysis of clustering algorithms and dimensionality reduction techniques (Saul, Weinberger, Ham, Sha, & Lee, 2006).On the other hand, spectral methods are being used to obtain information from massive data, focusing on the concept of the eigenvalue of a matrix, that is, a linear operator.See, for example (Chen, Chi, Fan, & Ma, 2021), The property (bz) allows us to discover various properties of the spectrum of an operator and was introduced by K. Ouidren and H. Zariouh in (Ben Ouidren & Zariouh, 2021) as a new variant of the classical Weyl's theorem; they further show that it is an extension of the classical a-Browder theorem.The study of this property leads to discovery the relationship that the upper semi-Fredholm spectrum has with the Browder-type spectra, and these are related to the upper semi-Weyl spectrum in (Aponte, Mac� ıas, Sanabria, & Soto, 2020), thus yielding various results through the local spectral theory that allow the development of the Fredholm type operators theory, for instance, in (Ben Ouidren & Zariouh, 2022) is showing that property ðbzÞ is equivalent to the localized SVEP checked outside the upper semi-Fredholm spectrum, in (Ben Ouidren, Ouahab, & Zariouh, 2023) and (Ben Ouidren & Zariouh, 2021), this property, (bz), has been extended to define new spectral properties, from which the coincidence between other classical spectra is obtained.In addition, in (Aponte, Mac� ıas, Sanabria, & Soto, 2021) the property (bz) is transmitted from an invertible Drazin operator to its reverse Drazin.
We see that this property (bz) has been developing, so it is interesting: to describe the spectral structure of an operator that verifies the property (bz) to obtain new relations with those given by the Weyl or Browder-type properties, to make simplifications in some calculations though restriction operator to a proper subspace, to study the transmission of this property (bz) to the tensor product of two factors that also verify it.
The purpose of this work is to deepen the operator theory, continuing the study of the property (bz), using several techniques of the local spectral theory.Specifically in: Section 3. We see that property (bz) may be characterized in several ways; in particular, to describe the spectral structure of the operators that verify it, we characterize it by means of the quasi-nilpotent part, the hyper-range, and the hyper-kernel of an operator.Also, an interest characterization through the concept of interior-point will allow proving that the set of operators that verify the property (bz) is closed.
Section 4. In (Ben Ouidren & Zariouh, 2021) the properties ðgbzÞ, ðW P a 00 Þ and ðgW P a 00 Þ are introduced, here we study new relations between the property ðbzÞ and those other three properties in order to obtain the conditions that give the equivalences among the four properties.
Section 5. We will investigate the property (bz) under a closed-proper subspace to simplify by its restriction on this subspace the computations of the operator in the space and thus obtain the properties enjoyed by the operator if its restriction operator verifies the property ðbzÞ: Section 6.We analyze the sufficient conditions that allow to transfer of the property (bz) of two tensor factors T and S to their tensor product T � S: Section 7. We draw some conclusions.

Definitions and basic results
In this section, C is the space of complex numbers.For T 2 LðXÞ, the Banach algebra of all bounded linear operators on a complex Banach space X, we put by aðTÞ the dimension of ker T (the Kernel of T), by bðTÞ the co-dimension of TðXÞ (the range of T), pðTÞ the ascent of T and qðTÞ the descent of T, and the spectrum of T defined as; SðTÞ ¼ f⋋ 2 C : ⋋I − T is not invertibleg: For the classical and well-known spectra, we give the following notations; S s ðTÞ, S e ðTÞ, S uf ðTÞ, S p ðTÞ, S ubf ðTÞ, S bf ðTÞ, S a ðTÞ, S w ðTÞ, S uw ðTÞ, S lw ðTÞ, S ubw ðTÞ, S bw ðTÞ, S ub ðTÞ, S b ðTÞ, S lb ðTÞ, S d ðTÞ, S ld ðTÞ, indicating the spectrum of surjective, Fredholm, upper semi-Fredholm, point, upper semi B-Fredholm, B-Fredholm, approximate point, Weyl, upper semi-Weyl, lower semi-Weyl, upper semi B-Weyl, B-Weyl, upper semi-Browder, Browder, lower semi-Browder, Drazin invertible, left Drazin invertible, respectively.For more details, see (Aiena, 2018).
The quasi-nilpotent part is given for the subspace hyper-range is ðTÞ n ðXÞ, and the subspace hyper-kernel is ker ðTÞ n : The boundary of the spectrum is always contained in the approximate point spectrum; see [ (Aiena, 2018), Theorem 1.12].The B-Browder spectrum coincides with the Drazin invertible spectrum, and the upper semi-B-Browder spectrum coincides with the left Drazin invertible spectrum.
The dual of X is X � :¼ LðX, CÞ: For T � 2 LðX � Þ, we denote the classical dual operator of T, defined by ðT � f ÞðXÞ :¼ f ðTxÞ for all x 2 X, f 2 X � : By HðSðTÞÞ, we denote the set of all analytic functions defined in an open neighbourhood of SðTÞ, and for f 2 HðSðTÞÞ, we define f ðTÞ as in the Riesz functional calculus.
We refer to (Aiena, 2018) for further details on notation and terminologies.
Definition 2.1 (Finch, 1975).T 2 LðXÞ possesses the SVEP in every isolated point of the spectrum SðTÞ and at every point of the resolvent qðTÞ ¼ C n SðTÞ, thus, in the border points of SðTÞ: Moreover, by [(Aiena, 2004), Theorem 3.8], we get that pð⋋I − TÞ < 1 ) T possesses the SVEP in ⋋, (1) and qð⋋I − TÞ < 1 ) T � possesses the SVEP in ⋋: (2) It is easily seen, from definition 2.1, that S a ðTÞ does not cluster at ⋋ and S s ðTÞ does not cluster at ⋋ ) T � possesses the SVEP in ⋋: (4) Note that by [ (Aiena, 2004), Theorem 2.31], we have that H 0 ð⋋I − TÞ is closed ) T possesses the SVEP in ⋋: (5) We consider for T 2 LðXÞ the set: NðTÞ ¼ f⋋ 2 C : T does not possess the SVEP in ⋋g: Clearly, NðTÞ is contained in the interior of the spectrum; according to the classical identity theorem for analytic functions, it follows that NðTÞ is open.Thus, if T possesses the SVEP in all ⋋ 2 Dð⋋ 0 , eÞ n f⋋ 0 g, where Dð⋋ 0 , eÞ is an open disc centered at ⋋ 0 , then T also possesses the SVEP in ⋋ 0 : Remark 2.2.(1)-( 5) are equivalences, whenever ⋋I − T is a quasi-Fredholm operator, see (Aiena, 2007), and in particular, when it is semi-Fredholm, semi B-Fredholm, left drazin invertible or right Drazin invertible.

Further characterizations of property (bz)
In this part, we give some characterizations of the property ðbzÞ, or equivalently the property ðgbzÞ, see (Ben Ouidren & Zariouh, 2021), via local spectral theory methods; in particular, T 2 LðXÞ verifies the property ðbzÞ, if and only if, T possesses the SVEP in the exterior of the upper semi-Fredholm spectrum.Among other results, we show that the set of operators that verify the property ðbzÞ contains all its limit points; that is, it is closed.Given T 2 LðXÞ, we define: � P a 0 ðTÞ : ¼ f⋋ 2 isoðS a ðTÞ Þ : 0 < að⋋I − TÞg: � P a 00 ðTÞ :¼ f⋋ 2 isoðS a ðTÞÞ : 0 < að⋋I − TÞ < 1g: � d uf ðTÞ :¼ S a ðTÞ n S uf ðTÞ: � d ubf ðTÞ :¼ S a ðTÞ n S ubf ðTÞ: � P a 0 ðTÞ :¼ S a ðTÞ n S ld ðTÞ: � P a 00 ðTÞ :¼ S a ðTÞ n S ub ðTÞ: Remark 3.1.For T 2 LðXÞ, P a 0 ðTÞ is the set of all left poles of T, and P a 00 ðTÞ the set of all left poles of T having finite rank.Thus, P a 00 ðTÞ � P a 0 ðTÞ: If ⋋ 2 P a 0 ðTÞ then ⋋I − T is left Drazin invertible and so pð⋋I − TÞ < 1: Then, ⋋I − T has topological uniform descent (see (Grabiner, 1982), for definition and details), of which by [ (Aiena & Sanabria, 2008) For T 2 L bz ðXÞ, the difference between the spectrum and the upper semi-Fredholm spectrum is contained in the surjective spectrum.Conversely, assume int ðd uf ðTÞÞ ¼ ;: Let ⋋ 0 2 d uf ðTÞ, if T does not possesses the SVEP at ⋋ 0 , then T does not possess the SVEP for all ⋋ 2 Dð⋋ 0 , e 0 Þ, for some e 0 > 0, since NðTÞ is an open set.Also, the set of operators upper semi-Fredholm is open, so there exists e 1 > 0 such that for all ⋋ 2 Dð⋋ 0 , e 1 Þ, ⋋I − T is upper semi-Fredholm.Now, if e ¼ minfe 0 , e 1 g then for every ⋋ 2 Dð⋋ 0 , eÞ, ⋋I − T has closed range, and so ⋋I − T is not injective, otherwise, ⋋I − T is bounded below and so T possesses the SVEP at ⋋, a contradiction.Consequently, Dð⋋ 0 , eÞ � d uf ðTÞ, but this is impossible.We have to by hence, T possesses the SVEP in ⋋ 0 , and by Remark 2.2, it turns out that pð⋋ 0 I − TÞ < 1, and so ⋋ 0 2 P a 00 ðTÞ: We deduce that d uf ðTÞ � P a 00 ðTÞ, so T 2 L bz ðXÞ: ðiÞ ) ðiiiÞ It can be deduced from Remark 3.1.ðivÞ ) ðiiiÞ ) ðiiÞ From the definition of these sets, the result is clear.
To close this section, we prove that the set of operators verifying the property (bz) is closed in LðXÞ: 2S uf ðT n Þ, for all n � N 0 : Since ⋋ 0 2 S a ðTÞ, so 0 < að⋋ 0 I − TÞ, by Remark 2.3 it turns out that dðker ð⋋ 0 I − T n Þ, ker ð⋋ 0 I − TÞÞ !0 as n ! 1, and there exists N 1 2 N such that ⋋ 0 2 S a ðT n Þ, for all n � N 1 : Thus, there exists N 2 2 N, such that int ðd uf ðTÞÞ � int ðd uf ðT n ÞÞ, for all n � N 2 : By hypothesis T N 2 2 L bz ðXÞ, thus by Theorem 3.5, result that int ðd uf ðT N 2 ÞÞ ¼ ;: This implies int ðd uf ðTÞÞ ¼ ;: Therefore, by Theorem 3.5, we have that T 2 L bz ðXÞ: ￭ Note that if T n is a sequence of operators on LðXÞ, satisfying the hypothesis of Theorem 3.11, and f 2 HðSðTÞÞ: So by Theorem 3.7, it follows that f ðT n Þ satisfies property (bz), so by Theorem 3.11, it turns out that f ðTÞ verifies property (bz).

Some relations between properties (bz), W P a
00 and gW P a 00 It is well known that ðgW P a 00 Þ ) ðW P a 00 Þ ) ðbzÞ: Now, in this section for T 2 LðXÞ verifying property ðbzÞ, we look for conditions so that T satisfies the property ðW P a 00 Þ, equally the property ðgW P a 00 Þ: Thus, in the latter case it turns out that ðgW P a 00 Þ () ðW P a 00 Þ () ðbzÞ: Equivalence that allows correlating some of the results of the previous section.The results obtained in this section will be applied in the following two sections.The relationship between these properties and the a-Weyl's theorem has already been studied.In particular, we have the following result.ðXÞ: (ii) It follows from hypothesis that ensure T is an a-polaroid operator which verifies property (bz).See part (i). ￭ The following result is very important in order to obtain many applications.
Corollary 4.6.Let T 2 LðXÞ an a-polaroid operator verifying one of the conditions of Theorem 3.5 or 3.6, then T 2 L gW P a 00 ðXÞ: Generalizing the previous corollary, we have the following result.
Corollary 4.7.Let T 2 LðXÞ an a-polaroid operator verifying one of the conditions of Theorem 3.5 or 3.6.If f 2 H nc ðSðTÞÞ, then f ðTÞ verifies property ðgW P a 00 Þ: Proof.By Theorem 3.7, we obtain that f ðTÞ 2 L bz ðXÞ: Moreover, by [ (Aiena, Aponte, & Balzan, 2010), Lemma 3.11], it turns out that f ðTÞ is an apolaroid operator.Therefore, by Theorem 4.5 it follows that f ðTÞ verifies the property ðgW P a 00 Þ: ￭ Recall that the a-polaroid condition implies the left polaroid condition.Now, the properties (bz) and ðgW P a 00 Þ are equivalent for operators that are both left polaroid and a-isoloid.Indeed, we have the following result.
Theorem 4.8.If T 2 L bz ðXÞ is an a-isoloid and left polaroid operator, then T 2 L gW P a 00 ðXÞ: Proof.Since T 2 L bz ðXÞ so T 2 L gbz ðXÞ, whereby d ubf ðTÞ ¼ S a ðTÞ n S ld ðTÞ: Thus, d ubf ðTÞ � P a 0 ðTÞ, because T is an a-isoloid operator.Note that S ubf ðTÞ � S ubw ðTÞ � S ld ðTÞ, and from T is an left polaroid operator, then we deduce that P a 0 ðTÞ � d ubf ðTÞ: Therefore, P a 0 ðTÞ ¼ d ubf ðTÞ and T 2 L gW P a 00 ðXÞ: Example 4.9.Every multiplier T of a semi-simple commutative Banach algebra A, is H(1), see (Aiena & Villafañe, 2005).Thus, as in the Example 3.9, we have that T 2 L bz ðXÞ: Note that T is a polaroid operator.Also, if A is regular and Tauberian, then by [ (Aiena, 2004), Corollary 5.88] result that SðTÞ ¼ S a ðTÞ: Thus, T is an a-polaroid operator.Now, if f 2 H nc ðSðTÞÞ, then by Corollary 4.7, we obtain that f ðTÞ verifies property ðgW P a 00 Þ: Then, the Theorems 3.5, 3.6 and 3.7 apply for f ðTÞ:

The property (bz) and proper subspaces
Let W a proper closed subspace of X, and consider the set PðX, WÞ ¼ fT 2 LðXÞ : TðWÞ � W, T n 0 ðXÞ � W, n 0 � 1g: Let T 2 PðX, WÞ, T W denotes the restriction of T over the T-invariant subspace W of X: So in this section, we characterize the properties (bz) and ðgW P a 00 Þ through the operator T W : Now, following [ (Aiena, 2004), Theorem 1.42], it is possible obtain a closed subspace T 1 ðXÞ ¼ KðTÞ, when T is a semi-Fredholm operator.
In (Carpintero, Guti� errez, Rosas, & Sanabria, 2020), for T W several spectra derived from the classical Fredholm theory are studied, and the relationship between such spectra with the corresponding spectra of T 2 LðXÞ is studied.In particular, we have the following theorem.
Corollary 5.5.Let T 2 LðXÞ be a semi-Fredholm operator with ascent or descent not finite.If T verifies one of the statements of Theorem 3.5 or 3.6, then there exists a proper closed subspace W of X such that T W 2 L bz ðXÞ: Proof.By hypothesis T has ascent or descent not finite, so that T is not surjective, whereby W ¼ T 1 ðXÞ ¼ KðTÞ is a proper closed subspace of X: Note that T 2 PðX, WÞ: Also, by Theorem 3.5 or 3.6, we get that T 2 L bz ðXÞ: Thus, by Theorem 5.3 it turns out that T W 2 L bz ðXÞ: ￭ Let T 2 PðX, WÞ an a-polaroid operator.If the operator T W is not an upper semi-Fredholm operator, then the property ðgW P a 00 Þ is transmitted from T to T W and vice-versa.
Theorem 5.6.Let W be a proper closed subspace of X and T 2 PðX, WÞ an a-polaroid operator such that 0 2 S uf ðT W Þ. Then, T 2 L gW P a 00 ðXÞ if and only if T W verifies property ðgW P a 00 Þ: Proof.Directly.Suppose that T 2 L gW P a 00 ðXÞ, thus by Theorem 4.5, T 2 L bz ðXÞ, and then using Theorem 5.2, we get that T W 2 L bz ðXÞ: Note that S a ðT W Þ ¼ S a ðTÞ, and as T is an a-polaroid operator, so by parts (ii) and (v) of Theorem 5.1, we deduce that T W is an a-polaroid operator.Hence, by Theorem 4.5, we obtain that T W verifies property ðgW P a 00 Þ: Conversely, if T W verifies property ðgW P a 00 Þ, then T W 2 L bz ðXÞ: Thus, by Theorem 5.2, we obtain that T 2 L bz ðXÞ and as T is an a-polaroid operator, so by Theorem 4.5, we have that T 2 L gW P a 00 ðXÞ:

The property (bz) under tensor product
Given two Banach spaces X and Y: So, X � Y denote the completion (in some reasonable cross norm) of the tensor product of X with Y: Also, T � S 2 LðX � YÞ denote the tensor product of T 2 LðXÞ with S 2 LðYÞ: There are studies on the conditions that allow transferring some spectral properties of two factors, T and S, to the tensor product T � S; for example, the properties, called (gaz) and (Bv), are studied under the tensor product, see (Aponte, Jayanthi, Quiroz, & Vasanthakumar, 2022) and (Aponte, Jayanthi, et al., 2022), respectively.
In this section, we see how the property (bz) is transmitted, without conditions, from two tensor The operator T 2 LðXÞ possesses the single-valued extension property in ⋋ 0 2 C (abbreviated SVEP in ⋋ 0 ) if for every open disc D with ⋋ 0 2 D, the only analytic function f : D !X which satisfies the equation ð⋋I − TÞf ð⋋Þ ¼ 0 for all ⋋ 2 D is the function f � 0: An operator T 2 LðXÞ possesses the SVEP if T possesses the SVEP in every point ⋋ 2 C: