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 $\mathbb{X},$ 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.

Keywords:

• Property (bz)
• semi-Fredholm operator
• tensor product
• $\mathit{SVEP}$

2020 Mathematics Subject Classification:

• Primary 47A10
• 47A11
• Secondary 47A53
• 47A55

1. 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, Macías, 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 $(bz)$ is equivalent to the localized $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ 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, Macías, 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 $(\mathit{gbz}),(W_{{\Pi }_{00}^a})$ and $(g{W}_{{\Pi }_{00}^a})$ are introduced, here we study new relations between the property $(bz)$ 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 $(bz).$

Section 6. We analyze the sufficient conditions that allow to transfer of the property (bz) of two tensor factors $\mathbb{T}$ and $\mathbb{S}$ to their tensor product $\mathbb{T}\otimes \mathbb{S}.$

Section 7. We draw some conclusions.

2. Definitions and basic results

In this section, $\mathbb{C}$ is the space of complex numbers. For $\mathbb{T}\in \mathbb{L}(\mathbb{X}),$ the Banach algebra of all bounded linear operators on a complex Banach space $\mathbb{X},$ we put by $\alpha (\mathbb{T})$ the dimension of $\text{ker}\mathbb{T}$ (the Kernel of $\mathbb{T}$), by $\beta (\mathbb{T})$ the co-dimension of $\mathbb{T}(\mathbb{X})$ (the range of $\mathbb{T}$), $p(\mathbb{T})$ the ascent of $\mathbb{T}$ and $q(\mathbb{T})$ the descent of $\mathbb{T},$ and the spectrum of $\mathbb{T}$ defined as; $\mathfrak{S}(\mathbb{T})=\{⋋\in \mathbb{C}:⋋\mathbb{I}-\mathbb{T}\text{is not invertible}\}.$

For the classical and well-known spectra, we give the following notations; ${\mathfrak{S}}_s(\mathbb{T}),{\mathfrak{S}}_e(\mathbb{T}),{\mathfrak{S}}_{uf}(\mathbb{T}),{\mathfrak{S}}_p(\mathbb{T}),{\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T}),{\mathfrak{S}}_{bf}(\mathbb{T}),{\mathfrak{S}}_a(\mathbb{T}),{\mathfrak{S}}_w(\mathbb{T}),$ ${\mathfrak{S}}_{uw}(\mathbb{T}),{\mathfrak{S}}_{lw}(\mathbb{T}),{\mathfrak{S}}_{\mathit{ubw}}(\mathbb{T}),{\mathfrak{S}}_{bw}(\mathbb{T}),{\mathfrak{S}}_{ub}(\mathbb{T}),$ ${\mathfrak{S}}_b(\mathbb{T}),$ ${\mathfrak{S}}_{lb}(\mathbb{T}),{\mathfrak{S}}_d(\mathbb{T}),{\mathfrak{S}}_{ld}(\mathbb{T}),$ 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 $${\mathbb{H}}_0(\mathbb{T}):=\{x\in \mathbb{X}:\underset{n\to \infty }{\lim }\Vert {(\mathbb{T})}^{n}x{\Vert }^{1/n}=0\},$$ the subspace hyper-range is $${(\mathbb{T})}^{\infty }(\mathbb{X}):=\underset{n=1}{\overset{\infty }{\cap }}{(\mathbb{T})}^n(\mathbb{X}),$$ and the subspace hyper-kernel is $${\mathcal{N}}^{\infty }(\mathbb{T}):=\underset{n=1}{\overset{\infty }{\cup }}\ker {(\mathbb{T})}^n.$$

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 $\mathbb{X}$ is ${\mathbb{X}}^{\ast }:=\mathbb{L}(\mathbb{X},\mathbb{C}).$ For ${\mathbb{T}}^{\ast }\in \mathbb{L}({\mathbb{X}}^{\ast }),$ we denote the classical dual operator of $\mathbb{T},$ defined by $$({\mathbb{T}}^{\ast }f)(\mathbb{X}):=f(\mathbb{T}x)\hspace{1em}\text{for all}x\in \mathbb{X}, f\in {\mathbb{X}}^{\ast }.$$

By $\mathcal{H}(\mathfrak{S}(\mathbb{T})),$ we denote the set of all analytic functions defined in an open neighbourhood of $\mathfrak{S}(\mathbb{T}),$ and for $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T})),$ we define $f(\mathbb{T})$ as in the Riesz functional calculus.

We refer to (Aiena, 2018) for further details on notation and terminologies.

Definition 2.1

(Finch, 1975). The operator $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ possesses the single-valued extension property in ${⋋}_0\in \mathbb{C}$ (abbreviated $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in ${⋋}_0$) if for every open disc $\mathbb{D}$ with ${⋋}_0\in \mathbb{D},$ the only analytic function $f:\mathbb{D}\to \mathbb{X}$ which satisfies the equation $(⋋\mathbb{I}-\mathbb{T})f(⋋)=0$ for all $⋋\in \mathbb{D}$ is the function $f\equiv 0.$

An operator $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ if $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in every point $⋋\in \mathbb{C}.$

$\mathbb{T}\in \mathbb{L}(\mathbb{X})$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in every isolated point of the spectrum $\mathfrak{S}(\mathbb{T})$ and at every point of the resolvent $\rho (\mathbb{T})=\mathbb{C}\setminus \mathfrak{S}(\mathbb{T}),$ thus, in the border points of $\mathfrak{S}(\mathbb{T}).$ Moreover, by [(Aiena, 2004), Theorem 3.8], we get that (1) $$p(⋋\mathbb{I}-\mathbb{T})<\infty \Rightarrow \mathbb{T}\text{possesses the }\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}\text{ in }⋋,$$(1) and (2) $$q(⋋\mathbb{I}-\mathbb{T})<\infty \Rightarrow {\mathbb{T}}^{*}\text{possesses the }\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}\text{ in}⋋.$$(2)

It is easily seen, from definition 2.1, that (3) $${\mathfrak{S}}_{\mathrm{a}}(\mathbb{T})\text{ does not cluster at}⋋\Rightarrow \mathbb{T}\text{possesses the }\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}\text{ in}⋋,$$(3) and (4) $${\mathfrak{S}}_{\mathrm{s}}(\mathbb{T})\text{does not cluster at}⋋\Rightarrow {\mathbb{T}}^{*}\text{ possesses the }\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}\text{ in}⋋.$$(4)

Note that by [(Aiena, 2004), Theorem 2.31], we have that (5) $${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})\text{is closed}\Rightarrow \mathbb{T}\text{possesses the }\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}\text{ in}⋋.$$(5)

We consider for $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ the set: $$\mathit{\Xi }(\mathbb{T})=\{⋋\in \mathbb{C}:\mathbb{T}\mathit{does}\mathit{not}\mathit{possess}\mathit{the}\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}in⋋\}.$$

Clearly, $\mathit{\Xi }(\mathbb{T})$ is contained in the interior of the spectrum; according to the classical identity theorem for analytic functions, it follows that $\mathit{\Xi }(\mathbb{T})$ is open. Thus, if $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in all $⋋\in D({⋋}_0,ϵ)\setminus \left\{{⋋}_0\right\},$ where $D({⋋}_0,ϵ)$ is an open disc centered at ${⋋}_0,$ then $\mathbb{T}$ also possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in ${⋋}_0.$

Remark 2.2.

(1)-(5)(5) $${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})\text{is closed}\Rightarrow \mathbb{T}\text{possesses the }\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}\text{ in}⋋.$$(5) are equivalences, whenever $⋋\mathbb{I}-\mathbb{T}$ 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.

If $\mathbb{M},\mathbb{N}$ closed linear sub-spaces of $\mathbb{X},$ we define $$\delta (\mathbb{M},\mathbb{N}):=\sup \{\text{dist} (u,\mathbb{N}):u\in \mathbb{M},\Vert u\Vert =1\},$$ in the case $\mathbb{M}\ne \left\{0\right\},$ otherwise set $\delta (\{0\},\mathbb{N})=0$ for any subspace $\mathbb{N}.$ Then, the gap between $\mathbb{M}$ and $\mathbb{N}$ is then defined by $$\widehat{\delta }(\mathbb{M},\mathbb{N}):=\max \{\delta (\mathbb{M},\mathbb{N}),\delta (\mathbb{N},\mathbb{M})\}.$$

The function $\widehat{\delta }$ is the gap metric on the set of all linear closed sub-spaces of $\mathbb{X},$ see [(Kato, 1966), §2, Chapter iv], the convergence ${\mathbb{M}}_n\to \mathbb{M}$ is clearly defined by $\widehat{\delta }({\mathbb{M}}_n,\mathbb{M})\to 0$ as $n\to \infty .$

Remark 2.3.

[(Müller, 2003), Chapter 10] If $\mathbb{T}(\mathbb{X})$ and ${\mathbb{T}}_n(\mathbb{X}),(1\le n),$ are closed in $\mathbb{X},$ and $\underset{n\to +\infty }{\lim } \parallel {\mathbb{T}}_n-\mathbb{T}\parallel =0,$ it turns out, $\widehat{\delta }(\text{ker} ({\mathbb{T}}_n),\text{ker} (\mathbb{T}))\to 0$ as $n\to \infty ,$ and $\widehat{\delta }({\mathbb{T}}_n(\mathbb{X}), \mathbb{T}(\mathbb{X}))\to 0$ as $n\to \infty .$ Therefore, dim $(\text{ker} ({\mathbb{T}}_n))=$ dim $(\text{ker} (\mathbb{T}))$ and dim $({\mathbb{T}}_n(\mathbb{X}))=$ dim $(\mathbb{T}(\mathbb{X})),$ for all $n\ge N_0.$

In $\mathbb{C},$ we denote by $\text{iso}(\mathbb{A}),\text{acc} (\mathbb{A})$ and $\partial (\mathbb{A}),Cl(\mathbb{A}),\text{int} (\mathbb{A}),$ the isolate points, the accumulate points, the border points, the closure points, and interior points, respectively of $\mathbb{A}\subseteq \mathbb{C}.$

Recall that $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ is said to be a-isoloid if $\text{iso}({\mathfrak{S}}_a(\mathbb{T}))\subseteq {\mathfrak{S}}_p(\mathbb{T}).$ $\mathbb{T}$ verifies a-Browder’s theorem if ${\mathfrak{S}}_{uw}(\mathbb{T})={\mathfrak{S}}_{ub}(\mathbb{T}),$ and $\mathbb{T}$ verifies a-Weyl’s theorem if $${\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_{uw}(\mathbb{T})={\Pi }_{00}^a(\mathbb{T}).$$

3. Further characterizations of property (bz)

In this part, we give some characterizations of the property $(bz),$ or equivalently the property $(\mathit{gbz}),$ see (Ben Ouidren & Zariouh, 2021), via local spectral theory methods; in particular, $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ verifies the property $(bz),$ if and only if, $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in the exterior of the upper semi-Fredholm spectrum. Among other results, we show that the set of operators that verify the property $(bz)$ contains all its limit points; that is, it is closed.

Given $\mathbb{T}\in \mathbb{L}(\mathbb{X}),$ we define:

• ${\Pi }_0^a(\mathbb{T}):=\{⋋\in \text{iso}({\mathfrak{S}}_a(\mathbb{T})):0<\alpha (⋋\mathbb{I}-\mathbb{T})\}.$

• ${\Pi }_{00}^a(\mathbb{T}):=\{⋋\in \text{iso}({\mathfrak{S}}_a(\mathbb{T})):0<\alpha (⋋\mathbb{I}-\mathbb{T})<\infty \}.$

• ${\delta }_{uf}(\mathbb{T}):={\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_{uf}(\mathbb{T}).$

• ${\delta }_{\mathit{ubf}}(\mathbb{T}):={\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T}).$

• ${\mathcal{P}}_0^a(\mathbb{T}):={\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_{ld}(\mathbb{T}).$

• ${\mathcal{P}}_{00}^a(\mathbb{T}):={\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_{ub}(\mathbb{T}).$

Remark 3.1.

For $\mathbb{T}\in \mathbb{L}(\mathbb{X}),$ ${\mathcal{P}}_0^a(\mathbb{T})$ is the set of all left poles of $\mathbb{T},$ and ${\mathcal{P}}_{00}^a(\mathbb{T})$ the set of all left poles of $\mathbb{T}$ having finite rank. Thus, ${\mathcal{P}}_{00}^a(\mathbb{T})\subseteq {\mathcal{P}}_0^a(\mathbb{T}).$

If $⋋\in {\mathcal{P}}_0^a(\mathbb{T})$ then $⋋\mathbb{I}-\mathbb{T}$ is left Drazin invertible and so $p(⋋\mathbb{I}-\mathbb{T})<\infty .$ Then, $⋋\mathbb{I}-\mathbb{T}$ has topological uniform descent (see (Grabiner, 1982), for definition and details), of which by [(Aiena & Sanabria, 2008), Corollary 4.8], $⋋\mathbb{I}-\mathbb{T}$ is bounded below in a punctured disc centered at $⋋,$ so that $⋋\in \text{iso}({\mathfrak{S}}_a(\mathbb{T})).$ Therefore, ${\mathcal{P}}_0^a(\mathbb{T})\subseteq \mathit{iso}({\mathfrak{S}}_a(\mathbb{T})).$ Consequently, $\mathit{int} ({\mathcal{P}}_{00}^a(\mathbb{T}))=\mathit{int} ({\mathcal{P}}_0^a(\mathbb{T}))=\varnothing .$

Note that ${\mathcal{P}}_{00}^a(\mathbb{T})\subseteq {\delta }_{uf}(\mathbb{T}),$ and as ${\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T})\subseteq {\mathfrak{S}}_{ld}(\mathbb{T})$ so ${\mathcal{P}}_0^a(\mathbb{T})\subseteq {\delta }_{\mathit{ubf}}(\mathbb{T}).$

Definition 3.2

(Ben Ouidren & Zariouh, 2021). It is said that an operator $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ verifies:

• property (bz) if ${\delta }_{uf}(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T}).$

• property (gbz) if ${\delta }_{\mathit{ubf}}(\mathbb{T})={\mathcal{P}}_0^a(\mathbb{T}).$

• property $(W_{{\Pi }_{00}^a})$ if ${\delta }_{uf}(\mathbb{T})={\Pi }_{00}^a(\mathbb{T}).$

• property $(g{W}_{{\Pi }_{00}^a})$ if ${\delta }_{\mathit{ubf}}(\mathbb{T})={\Pi }_0^a(\mathbb{T}).$

Let us define the following sets to be used in the following.

• ${\mathbb{L}}_{bz}(\mathbb{X}):=\{\mathbb{T}\in \mathbb{L}(\mathbb{X}): \mathbb{T}\text{ satisfies property }(bz)\}.$

• ${\mathbb{L}}_{\mathit{gbz}}(\mathbb{X}):=\{\mathbb{T}\in \mathbb{L}(\mathbb{X}): \mathbb{T}\text{ satisfies property }(\mathit{gbz})\}.$

• ${\mathbb{L}}_{W_{{\Pi }_{00}^a}}(\mathbb{X}):=\{\mathbb{T}\in \mathbb{L}(\mathbb{X}): \mathbb{T}\text{ satisfies property}(W_{{\Pi }_{00}^a})\}.$

• ${\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}):=\{\mathbb{T}\in \mathbb{L}(\mathbb{X}): \mathbb{T}\text{ satisfies property}(g{W}_{{\Pi }_{00}^a})\}.$

Example 3.3.

Consider the projection operator $\mathbb{P}\in \mathcal{L}({\mathcal{l}}^2(\mathbb{N})),$ defined by $$\mathbb{P}(x_1,x_2,\dots )=(0,x_2,x_3,\dots ).$$

Note that $\mathfrak{S}(\mathbb{P})={\mathfrak{S}}_a(\mathbb{P})=\{0,1\},{\mathfrak{S}}_{uf}(\mathbb{P})={\mathfrak{S}}_{ub}(\mathbb{P})=\left\{1\right\}$ and ${\Pi }_{00}^a(\mathbb{P})=\left\{0\right\}.$ So, $\mathbb{P}\in {\mathbb{L}}_{W_{{\Pi }_{00}^a}}(\mathbb{X})$ and $\mathbb{P}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Where $\mathbb{X}={\mathcal{l}}^2(\mathbb{N}).$

For $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ the difference between the spectrum and the upper semi-Fredholm spectrum is contained in the surjective spectrum.

Theorem 3.4.

If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$, then $\mathfrak{S}(\mathbb{T})={\mathfrak{S}}_{uf}(\mathbb{T})\cup {\mathfrak{S}}_s(\mathbb{T}).$

Proof.

It is well known that $\mathfrak{S}(\mathbb{T})={\mathfrak{S}}_s(\mathbb{T})\cup {\mathfrak{S}}_a(\mathbb{T})=\Xi (\mathbb{T})\cup {\mathfrak{S}}_{\mathrm{s}}(\mathbb{T}).$ Since $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ it follows that, for $⋋\notin {\mathfrak{S}}_{uf}(\mathbb{T})$ the operator $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋.$ Thus, we obtain that ${\delta }_{uf}(\mathbb{T})\subseteq {\mathfrak{S}}_s(\mathbb{T}),$ thus ${\mathfrak{S}}_a(\mathbb{T})\subseteq {\mathfrak{S}}_{uf}(\mathbb{T})\cup {\mathfrak{S}}_s(\mathbb{T}).$ Therefore, $\mathfrak{S}(\mathbb{T})={\mathfrak{S}}_{uf}(\mathbb{T})\cup {\mathfrak{S}}_s(\mathbb{T}).$ ￭

The property (bz) can be characterized in several ways. In particular, the following theorem illustrates a part of the spectral structure of the operators belonging to ${\mathbb{L}}_{bz}(\mathbb{X}).$

Theorem 3.5.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X}).$ Then the following are equivalent:

1. $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X});$

2. $\text{int} ({\delta }_{uf}(\mathbb{T}))=\varnothing ;$

3. ${\delta }_{uf}(\mathbb{T})\subseteq \partial ({\mathfrak{S}}_a(\mathbb{T}));$

4. ${\delta }_{uf}(\mathbb{T})\subseteq \text{iso}({\mathfrak{S}}_a(\mathbb{T}));$

5. ${\delta }_{uf}(\mathbb{T})\cap \text{acc} ({\mathfrak{S}}_a(\mathbb{T}))=\varnothing ;$

6. ${\delta }_{uf}(\mathbb{T})\subseteq {\mathcal{P}}_0^a(\mathbb{T});$

7. ${\mathfrak{S}}_a(\mathbb{T})={\mathfrak{S}}_{uf}(\mathbb{T})\cup \text{iso}({\mathfrak{S}}_a(\mathbb{T}));$

8. ${\mathfrak{S}}_a(\mathbb{T})={\mathfrak{S}}_{uf}(\mathbb{T})\cup \partial ({\mathfrak{S}}_a(\mathbb{T}));$

9. ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{ub}(\mathbb{T});$

10. ${\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T})={\mathfrak{S}}_{ld}(\mathbb{T}).$

Proof.

$\text{(i)}\iff \mathrm{(}\text{ii}\mathrm{)}$ Directly, suppose that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ i.e., ${\delta }_{uf}(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T}),$ then by Remark 3.1 result that $\text{int} ({\delta }_{uf}(\mathbb{T}))=\varnothing .$

Conversely, assume $\text{int} ({\delta }_{uf}(\mathbb{T}))=\varnothing .$ Let ${⋋}_0\in {\delta }_{uf}(\mathbb{T}),$ if $\mathbb{T}$ does not possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ at ${⋋}_0,$ then $\mathbb{T}$ does not possess the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ for all $⋋\in D({⋋}_0,{ϵ}_0),$ for some ${ϵ}_0>0,$ since $\mathit{\Xi }(\mathbb{T})$ is an open set. Also, the set of operators upper semi-Fredholm is open, so there exists ${ϵ}_1>0$ such that for all $⋋\in D({⋋}_0,{ϵ}_1),⋋\mathbb{I}-\mathbb{T}$ is upper semi-Fredholm. Now, if $ϵ=\min \{{ϵ}_0,{ϵ}_1\}$ then for every $⋋\in D({⋋}_0,ϵ),⋋\mathbb{I}-\mathbb{T}$ has closed range, and so $⋋\mathbb{I}-\mathbb{T}$ is not injective, otherwise, $⋋\mathbb{I}-\mathbb{T}$ is bounded below and so $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ at $⋋,$ a contradiction. Consequently, $D({⋋}_0,ϵ)\subseteq {\delta }_{uf}(\mathbb{T}),$ but this is impossible. We have to by hence, $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in ${⋋}_0,$ and by Remark 2.2, it turns out that $p({⋋}_{0}I-T)<\infty ,$ and so ${⋋}_0\in {\mathcal{P}}_{00}^a(\mathbb{T}).$ We deduce that ${\delta }_{uf}(\mathbb{T})\subseteq {\mathcal{P}}_{00}^a(\mathbb{T}),$ so $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

$\text{(i)}\Rightarrow \mathrm{(}\text{iii}\mathrm{)}$ It can be deduced from Remark 3.1.

$\mathrm{(}\text{iv}\mathrm{)}\Rightarrow \mathrm{(}\text{iii}\mathrm{)}\Rightarrow \mathrm{(}\text{ii}\mathrm{)}$ From the definition of these sets, the result is clear.

$\text{(i)}\Rightarrow \mathrm{(}\text{iv}\mathrm{)}$ Suppose that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ i.e., ${\delta }_{uf}(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T}).$ Hence, we have to ${\delta }_{uf}(\mathbb{T})\subseteq \text{iso}({\mathfrak{S}}_a(\mathbb{T})).$

$\mathrm{(}\text{iv}\mathrm{)}\iff \text{(v)}$ It is immediate.

$\text{(i)}\iff \mathrm{(}\text{vi}\mathrm{)}$ Directly, it is clear, since ${\delta }_{uf}(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T})\subseteq {\mathcal{P}}_0^a(\mathbb{T}).$ Conversely, suppose ${\delta }_{uf}(\mathbb{T})\subseteq {\mathcal{P}}_0^a(\mathbb{T}),$ so ${\delta }_{uf}(\mathbb{T})\subseteq \text{iso}({\mathfrak{S}}_a(\mathbb{T})),$ which is the item $\mathrm{(}\text{iv}\mathrm{)}.$ But, $\mathrm{(}\text{iv}\mathrm{)}$ implies $\mathrm{(}\text{ii}\mathrm{)}$ and this implies $\text{(i)}.$ Hence, $\mathrm{(}\text{vi}\mathrm{)}$ implies $\text{(i)}.$

$\mathrm{(}\text{vii}\mathrm{)}\iff \mathrm{(}\text{iv}\mathrm{)}$ Directly, it is clear. Conversely, assume that ${\delta }_{uf}(\mathbb{T})\subseteq \text{iso}({\mathfrak{S}}_a(\mathbb{T})),$ whereby, $\text{iso}({\mathfrak{S}}_a(\mathbb{T}))=\text{iso}({\mathfrak{S}}_a(\mathbb{T}))\cup {\delta }_{uf}(\mathbb{T}).$ But ${\mathfrak{S}}_{uf}(\mathbb{T})\subseteq {\mathfrak{S}}_a(\mathbb{T}),$ so that $$\text{iso}({\mathfrak{S}}_a(\mathbb{T}))\cup {\mathfrak{S}}_{uf}(\mathbb{T})=(\text{iso}({\mathfrak{S}}_a(\mathbb{T}))\cup {\delta }_{uf}(\mathbb{T}))\cup {\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T}).$$

$\mathrm{(}\text{viii}\mathrm{)}\iff \mathrm{(}\text{iii}\mathrm{)}$ Directly, it is clear. Conversely, assume that ${\delta }_{uf}(\mathbb{T})\subseteq \partial ({\mathfrak{S}}_a(\mathbb{T})),$ since ${\mathfrak{S}}_{uf}(\mathbb{T})\subseteq {\mathfrak{S}}_a(\mathbb{T}),$ so; $$\partial ({\mathfrak{S}}_a(\mathbb{T}))\cup {\mathfrak{S}}_{uf}(\mathbb{T})=(\partial ({\mathfrak{S}}_a(\mathbb{T}))\cup {\delta }_{uf}(\mathbb{T}))\cup {\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T}).$$ $\text{(i)}\iff \mathrm{(}\text{ix}\mathrm{)}$ Directly, suppose that $⋋\notin {\mathfrak{S}}_{uf}(\mathbb{T}).$ Now, if $⋋\in {\mathfrak{S}}_a(\mathbb{T})$ it turns out that $⋋\notin {\mathfrak{S}}_{ub}(\mathbb{T}),$ or, if $⋋\notin {\mathfrak{S}}_a(\mathbb{T})$ so $p(⋋\mathbb{I}-\mathbb{T})=0$ and so $⋋\notin {\mathfrak{S}}_{ub}(\mathbb{T}).$ Hence, we deduce that ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{ub}(\mathbb{T}).$ The reverse direction is clear.

$\mathrm{(}\text{ix}\mathrm{)}\iff (\mathrm{x})$ Since the properties $(bz)$ and $(\mathit{gbz})$ are equivalent, the result is obtained by similar reasoning as in the previous test and taking into account that ${\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T})\subseteq {\mathfrak{S}}_{\mathit{ubw}}(\mathbb{T})\subseteq {\mathfrak{S}}_{ld}(\mathbb{T})\subseteq {\mathfrak{S}}_a(\mathbb{T}).$ ￭

The following theorem shows us other characterizations of the property (bz) through the quasi-nilpotent part, the hyper-range, and the hyper-kernel.

Theorem 3.6.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X})$. Then the following are equivalent:

1. $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$;

2. For $⋋\in {\delta }_{uf}(\mathbb{T})$, there is $v=v(⋋)$ such that ${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})=\ker {(⋋\mathbb{I}-\mathbb{T})}^v$;

3. For $⋋\in {\delta }_{uf}(\mathbb{T}),{\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})$ is closed;

4. For $⋋\in {\delta }_{uf}(\mathbb{T}),⋋\notin \text{int} ({\mathfrak{S}}_a(\mathbb{T}))$;

5. For $⋋\in {\delta }_{uf}(\mathbb{T}),{\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})\cap K(⋋\mathbb{I}-\mathbb{T})=\varnothing$;

6. For $⋋\in {\delta }_{uf}(\mathbb{T}),N^{\infty }(⋋\mathbb{I}-\mathbb{T})\cap T^{\infty }(⋋\mathbb{I}-\mathbb{T})=\varnothing$;

7. For $⋋\in {\delta }_{uf}(\mathbb{T})$, there is $d=d(⋋)$ such that $\mathbb{T}{|}_{{(⋋\mathbb{I}-\mathbb{T})}^d(\mathbb{X})}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋$, where the subspace ${(⋋\mathbb{I}-\mathbb{T})}^d(\mathbb{X})$ is provided with the operator range topology;

8. For $⋋\in {\delta }_{uf}(\mathbb{T})$, there is $d=d(⋋)$ such that $\mathbb{T}{|}_{{(⋋\mathbb{I}-\mathbb{T})}^d(\mathbb{X})}$ is bounded below, where the subspace ${(⋋\mathbb{I}-\mathbb{T})}^d(\mathbb{X})$ is provided with the operator range topology.

Proof.

$\text{(i)}\Rightarrow \mathrm{(}\text{ii}\mathrm{)}$ Since $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ each $⋋\in {\delta }_{uf}(\mathbb{T})$ is a left pole of $\mathbb{T}.$ We have, by [(Aiena, 2018), Theorem 2.97], exists $v=v(⋋)$ such that ${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})={\ker }^v(⋋\mathbb{I}-\mathbb{T}).$

$\mathrm{(}\text{ii}\mathrm{)}\Rightarrow \mathrm{(}\text{iii}\mathrm{)}$ It is clear.

$\mathrm{(}\text{iii}\mathrm{)}\Rightarrow \text{(i)}$ Let $⋋\in {\delta }_{uf}(\mathbb{T}),$ so by Remark 2.2, we get that $p(⋋\mathbb{I}-\mathbb{T})<\infty ,$ whereby $⋋\in {\mathcal{P}}_{00}^a(\mathbb{T}).$ We deduce that ${\delta }_{uf}(\mathbb{T})\subseteq {\mathcal{P}}_{00}^a(\mathbb{T}).$ Thus, $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

$\text{(i)}\iff \mathrm{(}\text{iv}\mathrm{)}$ It follows from the part $\mathrm{(}\text{iv}\mathrm{)}$ of Theorem 3.5.

$\text{(i)}\iff \text{(v)}$ Directly, if $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ then $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋\in {\delta }_{uf}(\mathbb{T}),$ so by [(Aiena, 2018), Theorem 2.60], we have that ${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})\cap K(⋋\mathbb{I}-\mathbb{T})=\varnothing .$

Conversely, if for all $⋋\in {\delta }_{uf}(\mathbb{T})$ is ${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})\cap K(⋋\mathbb{I}-\mathbb{T})=\varnothing ,$ then, again by [(Aiena, 2018), Theorem 2.60], $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋,$ and so by Remark 2.2, we have that $p(⋋\mathbb{I}-\mathbb{T})<\infty .$ Therefore, $⋋\in {\mathcal{P}}_{00}^a(\mathbb{T}),$ we deduce that ${\delta }_{uf}(\mathbb{T})\subseteq {\mathcal{P}}_{00}^a(\mathbb{T}),$ whereby $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

$\text{(i)}\iff \mathrm{(}\text{vi}\mathrm{)}$ Directly, if $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ then for each $⋋\in {\delta }_{uf}(\mathbb{T}),$ we have that $⋋\mathbb{I}-\mathbb{T}$ has topological uniform descent (see (Aiena, 2018) for the definition) and $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋.$ Thus, by [(Aiena, 2018), Theorem 2.97] it turns out that $N^{\infty }(⋋\mathbb{I}-\mathbb{T})\cap T^{\infty }(⋋\mathbb{I}-\mathbb{T})=\varnothing ,$ for each $⋋\in {\delta }_{uf}(\mathbb{T}).$

Conversely, if for all $⋋\in {\delta }_{uf}(\mathbb{T})$ is $N^{\infty }(⋋\mathbb{I}-\mathbb{T})\cap T^{\infty }(⋋\mathbb{I}-\mathbb{T})=\varnothing ,$ then $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ for each $⋋\in {\delta }_{uf}(\mathbb{T}),$ see [(Aiena, 2018), Theorem 2.97]. In this way, $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

$\text{(i)}\iff \mathrm{(}\text{vii}\mathrm{)}$ and $\text{(i)}\iff \mathrm{(}\text{viii}\mathrm{)},$ they are similar to $\text{(i)}\iff \mathrm{(}\text{vi}\mathrm{)}.$ ￭

Below we see that the property $(bz)$ is transferred from $\mathbb{T}$ to $f(\mathbb{T}).$

Theorem 3.7.

If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ and $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T}))$, then $f(\mathbb{T})\in {\mathbb{L}}_{bz}(\mathbb{X}).$

Proof.

Suppose that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ then by Theorem 3.5 part (ix), we have that, ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{ub}(\mathbb{T}).$ Let $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T})),$ so $f({\mathfrak{S}}_{uf}(\mathbb{T}))=f({\mathfrak{S}}_{ub}(\mathbb{T})),$ thus by applying [(Aiena, 2018), Theorem 3.109], it follows that ${\mathfrak{S}}_{uf}(f(\mathbb{T}))={\mathfrak{S}}_{ub}(f(\mathbb{T})).$ Therefore, $f(\mathbb{T})\in {\mathbb{L}}_{bz}(\mathbb{X}).$ ￭

In general ${\mathfrak{S}}_{uw}(f(\mathbb{T}))\subseteq f({\mathfrak{S}}_{uw}(\mathbb{T})).$ Now, we obtain:

Corollary 3.8.

If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$, then ${\mathfrak{S}}_{uw}(f(\mathbb{T}))=f({\mathfrak{S}}_{uw}(\mathbb{T})).$

Proof.

Let $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T}))$ so by Theorem 3.7, we get that $f(\mathbb{T})\in {\mathbb{L}}_{bz}(\mathbb{X}),$ whereby ${\mathfrak{S}}_{uf}(f(\mathbb{T}))={\mathfrak{S}}_{uw}(f(\mathbb{T}))={\mathfrak{S}}_{ub}(f(\mathbb{T})).$ Analogously, we have that ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{uw}(\mathbb{T})={\mathfrak{S}}_{ub}(\mathbb{T}),$ but f is a function and ${\mathfrak{S}}_{ub}(f(\mathbb{T}))=f({\mathfrak{S}}_{ub}(\mathbb{T})),$ it turns out that ${\mathfrak{S}}_{uw}(f(\mathbb{T}))=f({\mathfrak{S}}_{uw}(\mathbb{T})).$ ￭

Example 3.9.

$\mathbb{T}\in \mathbb{L}(\mathbb{X})$ belongs to the class H(p) if there exists $p:=p(⋋),$ such that ${\mathbb{H}}_0(⋋\mathbb{I}-\mathbb{T})=\mathit{ker}{(⋋\mathbb{I}-\mathbb{T})}^p,$ for all $⋋\in \mathbb{C}.$ Then, by Theorem 3.6 part (iii), it turns out that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Thus, we obtain that Theorems 3.5, 3.6 and 3.7 applies for $\mathbb{T}.$

Example 3.10.

Let $\mathbb{D}$ be the open unit disk, $1<p<\infty ,$ and the Hardy space ${\mathbb{H}}_p(\mathbb{D}).$ It is defined the Césaro operator by: $$(C_{p}f)(⋋):=\frac{1}{⋋}{\int }_0^{⋋}\frac{f(\mu )}{(1-\mu )}d\mu ,\hspace{1em}\text{for} \text{all} \mathrm{f}\in {\mathbb{H}}_{\mathrm{p}}(\mathbb{D}) \text{and} ⋋\in \mathbb{D}.$$

Now, $\mathfrak{S}(C_p)={\mathbb{D}}_{\frac{p}{2}},$ which is the closed disc centered at $\frac{p}{2}$ and with radius $\frac{p}{2},$ also ${\mathfrak{S}}_a(C_p)=\partial {\mathbb{D}}_{\frac{p}{2}},$ see (Miller, Miller, & Smith, 1998). Clearly C_p possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋\in \mathfrak{S}(C_p)\setminus {\mathfrak{S}}_a(C_p),$ in this case ${\mathfrak{S}}_a(C_p)$ is the border of $\mathfrak{S}(C_p).$ Therefore, C_p possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ and so verifies property (bz). As, $\text{iso}({\mathfrak{S}}_a(C_p))=\varnothing ,$ so by Theorem 3.5 part (vii), we have that ${\mathfrak{S}}_a(C_p)={\mathfrak{S}}_{uf}(C_p)=\partial {\mathbb{D}}_{\frac{p}{2}},$ and by part (iv), we have that ${\mathfrak{S}}_{ub}(C_p)=\partial {\mathbb{D}}_{\frac{p}{2}}.$ Actually the Theorems 3.5, 3.6 and 3.7 apply for C_p.

To close this section, we prove that the set of operators verifying the property (bz) is closed in $\mathbb{L}(\mathbb{X}).$

Theorem 3.11.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ and ${\mathbb{T}}_n\in \mathbb{L}(\mathbb{X})$ for each $n\ge 1$, such that lim${}_{n\to +\infty }\parallel {\mathbb{T}}_n-\mathbb{T}\parallel =0$. If for each $n\ge 1,{\mathbb{T}}_n\in {\mathbb{L}}_{bz}(\mathbb{X})$, then $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

Proof.

The set of upper semi-Fredholm operators is open in $\mathbb{L}(\mathbb{X}),$ and its elements have closed rank. Let ${⋋}_0\in {\delta }_{uf}(\mathbb{T}),$ by convergence of ${\mathbb{T}}_n$ to $\mathbb{T},$ there exists $N_0\in \mathbb{N}$ such that ${⋋}_0\notin {\mathfrak{S}}_{uf}({\mathbb{T}}_n),$ for all $n\ge N_0.$

Since ${⋋}_0\in {\mathfrak{S}}_a(\mathbb{T}),$ so $0<\alpha ({⋋}_0\mathbb{I}-\mathbb{T}),$ by Remark 2.3 it turns out that $\widehat{\delta }(\text{ker} ({⋋}_0\mathbb{I}-{\mathbb{T}}_n),\text{ker} ({⋋}_0\mathbb{I}-\mathbb{T}))\to 0$ as $n\to \infty ,$ and there exists $N_1\in \mathbb{N}$ such that ${⋋}_0\in {\mathfrak{S}}_a({\mathbb{T}}_n),$ for all $n\ge N_1.$ Thus, there exists $N_2\in \mathbb{N},$ such that $\text{int} \left({\delta }_{uf}\right(\mathbb{T}\left)\right)\subseteq \text{int} \left({\delta }_{uf}\right({\mathbb{T}}_n\left)\right),\text{ for all n}\ge {\mathrm{N}}_2.$

By hypothesis ${\mathbb{T}}_{N_2}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ thus by Theorem 3.5, result that $\text{int} ({\delta }_{uf}({\mathbb{T}}_{N_2}))=\varnothing .$ This implies $\text{int} ({\delta }_{uf}(\mathbb{T}))=\varnothing .$ Therefore, by Theorem 3.5, we have that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ ￭

Note that if ${\mathbb{T}}_n$ is a sequence of operators on $\mathbb{L}(\mathbb{X}),$ satisfying the hypothesis of Theorem 3.11, and $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T})).$ So by Theorem 3.7, it follows that $f({\mathbb{T}}_n)$ satisfies property (bz), so by Theorem 3.11, it turns out that $f(\mathbb{T})$ verifies property (bz).

4. Some relations between properties (bz), $(W_{{\Pi }_{00}^a})$ and $(g{W}_{{\Pi }_{00}^a})$

It is well known that $(g{W}_{{\Pi }_{00}^a})\Rightarrow (W_{{\Pi }_{00}^a})\Rightarrow (bz).$ Now, in this section for $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ verifying property $(bz),$ we look for conditions so that $\mathbb{T}$ satisfies the property $(W_{{\Pi }_{00}^a}),$ equally the property $(g{W}_{{\Pi }_{00}^a}).$ Thus, in the latter case it turns out that $(g{W}_{{\Pi }_{00}^a})\iff (W_{{\Pi }_{00}^a})\iff (bz).$ 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.

Theorem 4.1

(Ben Ouidren & Zariouh, 2022). Let $\mathbb{T}\in \mathbb{L}(\mathbb{X})$, then:

1. $\mathbb{T}\in {\mathbb{L}}_{W_{{\Pi }_{00}^a}}(\mathbb{X})$ if and only if $\mathbb{T}$ verifies a-Weyl’s theorem and ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{uw}(\mathbb{T}).$

2. $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X})$ if and only if $\mathbb{T}$ verifies generalized a-Weyl’s theorem and ${\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T})={\mathfrak{S}}_{\mathit{ubw}}(\mathbb{T}).$

The property (bz) attains the property $(W_{{\Pi }_{00}^a}),$ if each eigenvalue is a point of the upper semi-Fredholm resolvent. In effect.

Theorem 4.2.

If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ and ${\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_p(\mathbb{T})\subseteq {\delta }_{uf}(\mathbb{T})$, then $\mathbb{T}\in {\mathbb{L}}_{W_{{\Pi }_{00}^a}}(\mathbb{X}).$

Proof.

Assume that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ and ${\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_p(\mathbb{T})\subseteq {\delta }_{uf}(\mathbb{T}).$ Let $⋋\in {\Pi }_{00}^a(\mathbb{T})$ so $⋋\in {\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_p(\mathbb{T})$ and so $⋋\notin {\mathfrak{S}}_{uf}(\mathbb{T}).$ Now, as $⋋\in \text{iso}({\mathfrak{S}}_a(\mathbb{T}))$ so $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋,$ by Remark 2.2 it turns out that $⋋\notin {\mathfrak{S}}_{ub}(\mathbb{T}).$ Therefore ${\Pi }_{00}^a(\mathbb{T})\subseteq {\mathcal{P}}_{00}^a(\mathbb{T}),$ thus ${\Pi }_{00}^a(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T}),$ but $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ i.e., ${\delta }_{uf}(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T}).$ Therefore, $\mathbb{T}\in {\mathbb{L}}_{W_{{\Pi }_{00}^a}}(\mathbb{X}).$ ￭

The equivalence in Theorem 4.1 part (ii) is given if the approximate point spectrum has no isolated points and coincides with the upper semi-Fredholm spectrum.

Theorem 4.3.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ such that ${\delta }_{uf}(\mathbb{T})=\text{iso}({\mathfrak{S}}_a(\mathbb{T}))=\varnothing$, then $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$

Proof.

From ${\Pi }_{00}^a(\mathbb{T})=\varnothing ,$ we get that $\mathbb{T}$ verifies property $(W_{{\Pi }_{00}^a})$ and so property (bz), or equivalently property (gbz), i.e., ${\delta }_{\mathit{ubf}}(\mathbb{T})={\mathcal{P}}_0^a(\mathbb{T}).$ As, ${\mathcal{P}}_0^a(\mathbb{T})\subseteq \text{iso}({\mathfrak{S}}_a(\mathbb{T}))$ so ${\delta }_{\mathit{ubf}}(\mathbb{T})={\Pi }_0^a(\mathbb{T}).$ Therefore, $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$ ￭

Example 4.4.

Let $\mathbb{T}$ be the Césaro operator, thus by Example 3.10, we have that $\text{iso}({\mathfrak{S}}_a(\mathbb{T}))=\varnothing ,{\mathfrak{S}}_a(\mathbb{T})={\mathfrak{S}}_{uf}(\mathbb{T})=\partial {\mathbb{D}}_{\frac{p}{2}}.$ Note that ${\mathfrak{S}}_p(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T}),$ so by Theorem 4.2, we get that $\mathbb{T}$ verifies property $(W_{{\Pi }_{00}^a}).$ Actually by Theorem 4.3, $\mathbb{T}$ verifies property $(g{W}_{{\Pi }_{00}^a}).$

Next, we see that properties (bz) and $(g{W}_{{\Pi }_{00}^a})$ are equivalent for the a-polaroid operators.

Theorem 4.5.

$\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X})$ if:

1. $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ and $\mathbb{T}$ is a-polaroid.

2. $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),\mathfrak{S}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T})$ and $\mathbb{T}$ is polaroid.

Proof.

(i) If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ then $\mathbb{T}\in {\mathbb{L}}_{\mathit{gbz}}(\mathbb{X}),$ whence ${\delta }_{\mathit{ubf}}(\mathbb{T})={\mathcal{P}}_0^a(\mathbb{T}).$ Since $\mathbb{T}$ is a-polaroid, it follows that ${\mathcal{P}}_0^a(\mathbb{T})\subseteq {\Pi }_0^a(\mathbb{T}).$ On the other hand, if $⋋\in {\Pi }_0^a(\mathbb{T}),$ so $⋋$ is a polo of $\mathbb{T},$ in this way $⋋\notin {\mathfrak{S}}_d(\mathbb{T}),$ so that $⋋\notin {\mathfrak{S}}_{ld}(\mathbb{T}),$ whereby $⋋\in {\mathcal{P}}_0^a(\mathbb{T}).$ We conclude that ${\mathcal{P}}_0^a(\mathbb{T})={\Pi }_0^a(\mathbb{T}).$ Therefore, $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$

(ii) It follows from hypothesis that ensure $\mathbb{T}$ 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 $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ an a-polaroid operator verifying one of the conditions of Theorem 3.5 or 3.6, then $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$

Generalizing the previous corollary, we have the following result.

Corollary 4.7.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ an a-polaroid operator verifying one of the conditions of Theorem 3.5 or 3.6. If $f\in {\mathcal{H}}_{nc}(\mathfrak{S}(\mathbb{T}))$, then $f(\mathbb{T})$ verifies property $(g{W}_{{\Pi }_{00}^a}).$

Proof.

By Theorem 3.7, we obtain that $f(\mathbb{T})\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Moreover, by [(Aiena, Aponte, & Balzan, 2010), Lemma 3.11], it turns out that $f(\mathbb{T})$ is an a-polaroid operator. Therefore, by Theorem 4.5 it follows that $f(\mathbb{T})$ verifies the property $(g{W}_{{\Pi }_{00}^a}).$ ￭

Recall that the a-polaroid condition implies the left polaroid condition. Now, the properties (bz) and $(g{W}_{{\Pi }_{00}^a})$ are equivalent for operators that are both left polaroid and a-isoloid. Indeed, we have the following result.

Theorem 4.8.

If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ is an a-isoloid and left polaroid operator, then $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$

Proof.

Since $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ so $\mathbb{T}\in {\mathbb{L}}_{\mathit{gbz}}(\mathbb{X}),$ whereby ${\delta }_{\mathit{ubf}}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T})\setminus {\mathfrak{S}}_{ld}(\mathbb{T}).$ Thus, ${\delta }_{\mathit{ubf}}(\mathbb{T})\subseteq {\Pi }_0^a(\mathbb{T}),$ because $\mathbb{T}$ is an a-isoloid operator. Note that ${\mathfrak{S}}_{\mathit{ubf}}(\mathbb{T})\subseteq {\mathfrak{S}}_{\mathit{ubw}}(\mathbb{T})\subseteq {\mathfrak{S}}_{ld}(\mathbb{T}),$ and from $\mathbb{T}$ is an left polaroid operator, then we deduce that ${\Pi }_0^a(\mathbb{T})\subseteq {\delta }_{\mathit{ubf}}(\mathbb{T}).$ Therefore, ${\Pi }_0^a(\mathbb{T})={\delta }_{\mathit{ubf}}(\mathbb{T})$ and $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$

Example 4.9.

Every multiplier $\mathbb{T}$ 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 $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Note that $\mathbb{T}$ is a polaroid operator. Also, if A is regular and Tauberian, then by [(Aiena, 2004), Corollary 5.88] result that $\mathfrak{S}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T}).$ Thus, $\mathbb{T}$ is an a-polaroid operator. Now, if $f\in {\mathcal{H}}_{nc}(\mathfrak{S}(\mathbb{T})),$ then by Corollary 4.7, we obtain that $f(\mathbb{T})$ verifies property $(g{W}_{{\Pi }_{00}^a}).$ Then, the Theorems 3.5, 3.6 and 3.7 apply for $f(\mathbb{T}).$

5. The property (bz) and proper subspaces

Let $\mathbb{W}$a proper closed subspace of $\mathbb{X},$ and consider the set $$\mathcal{P}(\mathbb{X},\mathbb{W})=\{\mathbb{T}\in \mathbb{L}(\mathbb{X}):\mathbb{T}(\mathbb{W})\subseteq \mathbb{W},{\mathbb{T}}^{n_0}(\mathbb{X})\subseteq \mathbb{W},n_0\ge 1\}.$$

Let $\mathbb{T}\in \mathcal{P}(\mathbb{X},\mathbb{W}),{\mathbb{T}}_{\mathbb{W}}$ denotes the restriction of $\mathbb{T}$ over the $\mathbb{T}$-invariant subspace $\mathbb{W}$ of $\mathbb{X}.$ So in this section, we characterize the properties (bz) and $(g{W}_{{\Pi }_{00}^a})$ through the operator ${\mathbb{T}}_{\mathbb{W}}.$

Now, following [(Aiena, 2004), Theorem 1.42], it is possible obtain a closed subspace ${\mathbb{T}}^{\infty }(\mathbb{X})=K(\mathbb{T}),$ when $\mathbb{T}$ is a semi-Fredholm operator.

In (Carpintero, Gutiérrez, Rosas, & Sanabria, 2020), for ${\mathbb{T}}_{\mathbb{W}}$ several spectra derived from the classical Fredholm theory are studied, and the relationship between such spectra with the corresponding spectra of $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ is studied. In particular, we have the following theorem.

Theorem 5.1

(Carpintero et al., 2020). Let $\mathbb{W}$ be a proper closed subspace of $\mathbb{X}$ and $\mathbb{T}\in P(\mathbb{X},\mathbb{W})$. If $⋋\ne 0$, then:

1. $(⋋\mathbb{I}-\mathbb{T})(\mathbb{X})$ is closed in $\mathbb{X}$ if and only if $(⋋\mathbb{I}-{\mathbb{T}}_{\mathbb{W}})(\mathbb{W})$ is closed in $\mathbb{W}.$

2. ${(⋋\mathbb{I}-{\mathbb{T}}_{\mathbb{W}})}^n(\mathbb{W})={(⋋\mathbb{I}-\mathbb{T})}^n(\mathbb{X})\cap \mathbb{W}$ for each n.

3. $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋$ if and only if ${\mathbb{T}}_{\mathbb{W}}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ in $⋋.$

4. $\alpha (⋋\mathbb{I}-\mathbb{T})=\alpha (⋋\mathbb{I}-{\mathbb{T}}_{\mathbb{W}}).$

5. $p(⋋\mathbb{I}-\mathbb{T})=p(⋋\mathbb{I}-{\mathbb{T}}_{\mathbb{W}}).$

Let $\mathbb{T}\in \mathcal{P}(\mathbb{X},\mathbb{W}).$ If the operator ${\mathbb{T}}_{\mathbb{W}}$ is not an upper semi-Fredholm operator, then the property (bz) is transmitted from $\mathbb{T}$ to ${\mathbb{T}}_{\mathbb{W}}$ and vice-versa.

Theorem 5.2.

Let $\mathbb{T}\in \mathcal{P}(\mathbb{X},\mathbb{W})$ such that $0\in {\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}})$. Then, $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ if and only if ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

Proof.

Let $⋋\notin {\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}}),$ so $⋋\ne 0,$ whereby by parts (i) and (iv) of Theorem 5.1, we have that $(⋋\mathbb{I}-\mathbb{T})(\mathbb{X})$ is closed and $\alpha (⋋\mathbb{I}-\mathbb{T})<\infty ,$ thus $⋋\notin {\mathfrak{S}}_{uf}(\mathbb{T}).$ Thus, ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}}).$ Now; $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ if and only if $\mathbb{T}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ at each $⋋\notin {\mathfrak{S}}_{uf}(\mathbb{T}),$ equivalently by part (iii) of Theorem 5.1, ${\mathbb{T}}_{\mathbb{W}}$ possesses the $\mathcal{S}\mathcal{V}\mathcal{E}\mathcal{P}$ at each $⋋\notin {\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}})$ if and only if ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ ￭

In case that ${\mathbb{T}}_{\mathbb{W}}$ is an upper semi-Fredholm operator, i.e., $0\notin {\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}}),$ the above result is obtained if the operator $\mathbb{T}$ has infinite ascent or descent.

Theorem 5.3.

Let $\mathbb{T}\in P(\mathbb{X},\mathbb{W})$ such that $q(\mathbb{T})=\infty ,orp(\mathbb{T})=\infty .$ Then, $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ if and only if ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

Proof.

Since, $q(\mathbb{T})=\infty$ or $p(\mathbb{T})=\infty ,$ so by [(Carpintero et al., 2020), Theorem 4.1] it turns out that ${\mathfrak{S}}_a({\mathbb{T}}_{\mathbb{W}})={\mathfrak{S}}_a(\mathbb{T}),{\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}})$ and ${\mathfrak{S}}_{\text{ub}}(\mathbb{T})={\mathfrak{S}}_{\text{ub}}({\mathbb{T}}_{\mathbb{W}}).$ Then, the result follows from the definition of property (bz). ￭

Example 5.4.

Let $R\in {\mathcal{l}}^2(\mathbb{N})$ be the unilateral right shift given by $$R(x_1,x_2,\dots ):=(0,x_1,x_2,\dots ),\hspace{1em}\text{for all}(x_n)\in {\mathcal{l}}^2(\mathbb{N}).$$

Note that $0\notin {\mathfrak{S}}_{uf}(R),$ thus $\mathbb{W}=R^{\infty }({\mathcal{l}}^2(\mathbb{N}))$ is a proper closed subspace of ${\mathcal{l}}^2(\mathbb{N}).$ Clearly, $R\in \mathcal{P}({\mathcal{l}}^2(\mathbb{N}),\mathbb{W})$ and $q(R)=\infty .$ On the other hand, in [(Ben Ouidren & Zariouh, 2022), Example 2.2], we can see that R verifies property (bz). Therefore, by Theorem 5.3, $R_{\mathbb{W}}$ verifies property (bz).

Corollary 5.5.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ be a semi-Fredholm operator with ascent or descent not finite. If $\mathbb{T}$ verifies one of the statements of Theorem 3.5 or 3.6, then there exists a proper closed subspace $\mathbb{W}$ of $\mathbb{X}$ such that ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$

Proof.

By hypothesis $\mathbb{T}$ has ascent or descent not finite, so that $\mathbb{T}$ is not surjective, whereby $\mathbb{W}={\mathbb{T}}^{\infty }(\mathbb{X})=K(\mathbb{T})$ is a proper closed subspace of $\mathbb{X}.$ Note that $\mathbb{T}\in \mathcal{P}(\mathbb{X},\mathbb{W}).$ Also, by Theorem 3.5 or 3.6, we get that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Thus, by Theorem 5.3 it turns out that ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ ￭

Let $\mathbb{T}\in \mathcal{P}(\mathbb{X},\mathbb{W})$ an a-polaroid operator. If the operator ${\mathbb{T}}_{\mathbb{W}}$ is not an upper semi-Fredholm operator, then the property $(g{W}_{{\Pi }_{00}^a})$ is transmitted from $\mathbb{T}$ to ${\mathbb{T}}_{\mathbb{W}}$ and vice-versa.

Theorem 5.6.

Let $\mathbb{W}$ be a proper closed subspace of $\mathbb{X}$ and $\mathbb{T}\in P(\mathbb{X},\mathbb{W})$ an a-polaroid operator such that $0\in {\mathfrak{S}}_{uf}({\mathbb{T}}_{\mathbb{W}})$. Then, $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X})$ if and only if ${\mathbb{T}}_{\mathbb{W}}$ verifies property $(g{W}_{{\Pi }_{00}^a}).$

Proof.

Directly. Suppose that $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}),$ thus by Theorem 4.5, $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}),$ and then using Theorem 5.2, we get that ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Note that ${\mathfrak{S}}_a({\mathbb{T}}_{\mathbb{W}})={\mathfrak{S}}_a(\mathbb{T}),$ and as $\mathbb{T}$ is an a-polaroid operator, so by parts (ii) and (v) of Theorem 5.1, we deduce that ${\mathbb{T}}_{\mathbb{W}}$ is an a-polaroid operator. Hence, by Theorem 4.5, we obtain that ${\mathbb{T}}_{\mathbb{W}}$ verifies property $(g{W}_{{\Pi }_{00}^a}).$

Conversely, if ${\mathbb{T}}_{\mathbb{W}}$ verifies property $(g{W}_{{\Pi }_{00}^a}),$ then ${\mathbb{T}}_{\mathbb{W}}\in {\mathbb{L}}_{bz}(\mathbb{X}).$ Thus, by Theorem 5.2, we obtain that $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ and as $\mathbb{T}$ is an a-polaroid operator, so by Theorem 4.5, we have that $\mathbb{T}\in {\mathbb{L}}_{g{W}_{{\Pi }_{00}^a}}(\mathbb{X}).$

6. The property (bz) under tensor product

Given two Banach spaces $\mathbb{X}$ and $\mathbb{Y}.$ So, $\mathbb{X}\otimes \mathbb{Y}$ denote the completion (in some reasonable cross norm) of the tensor product of $\mathbb{X}$ with $\mathbb{Y}.$ Also, $\mathbb{T}\otimes \mathbb{S}\in \mathbb{L}(\mathbb{X}\otimes \mathbb{Y})$ denote the tensor product of $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ with $\mathbb{S}\in \mathbb{L}(\mathbb{Y}).$ There are studies on the conditions that allow transferring some spectral properties of two factors, $\mathbb{T}$ and $\mathbb{S},$ to the tensor product $\mathbb{T}\otimes \mathbb{S};$ 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 factors, $\mathbb{T}$ and $\mathbb{S},$ until its tensor product $\mathbb{T}\otimes \mathbb{S}.$ While the properties $(W_{{\Pi }_{00}^a})$ and $(g{W}_{{\Pi }_{00}^a})$ do require some conditions to perform the mentioned transmission.

In (Duggal, Djordjevic, & Kubrusly, 2010), for $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ and $\mathbb{S}\in \mathbb{L}(\mathbb{Y}),$ we see that the following properties are satisfied:

1. ${\mathfrak{S}}_a(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S}).$

2. ${\mathfrak{S}}_{uf}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{S}){\mathfrak{S}}_{uf}(\mathbb{T})\cup {\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{uf}(\mathbb{S}).$

3. $\text{iso}({\mathfrak{S}}_a(\mathbb{T}\otimes \mathbb{S}))\subseteq \text{iso}({\mathfrak{S}}_a(\mathbb{T}))\text{iso}({\mathfrak{S}}_a(\mathbb{S}))\cup \left\{0\right\}.$

4. ${\mathfrak{S}}_{ub}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{ub}(\mathbb{S})\cup {\mathfrak{S}}_{ub}(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S}).$

5. If $\mathbb{T},\mathbb{S}$ and $\mathbb{T}\otimes \mathbb{S}$ verify a-Browder’s theorem, then $⋋=\mu \nu \in {\mathcal{P}}_{00}^a(\mathbb{T}\otimes \mathbb{S})$ if $\mu \in {\mathcal{P}}_{00}^a(\mathbb{T})$ and $\nu \in {\mathcal{P}}_{00}^a(\mathbb{S}).$ Also, ${\mathfrak{S}}_{uw}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{uw}(\mathbb{S})\cup {\mathfrak{S}}_{uw}(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S}).$

6. $\mathbb{T}\otimes \mathbb{S}$ satisfies a-Weyl’s theorem if $\mathbb{T}$ and $\mathbb{S}$ are a-isoloid operator satisfying a-Weyl’s theorem and ${\mathfrak{S}}_{uw}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{uw}(\mathbb{S})\cup {\mathfrak{S}}_{uw}(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S}).$

In the Next result, we see how the property (bz) is transmitted directly from two tensor factors to their tensor product.

Theorem 6.1.

If $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X})$ and $\mathbb{S}\in {\mathbb{L}}_{bz}(\mathbb{Y})$. Then $\mathbb{T}\otimes \mathbb{S}$ satisfies property (bz).

Proof.

By hypothesis it is obtained that ${\mathfrak{S}}_{uf}(\mathbb{T})={\mathfrak{S}}_{ub}(\mathbb{T})$ and ${\mathfrak{S}}_{uf}(\mathbb{S})={\mathfrak{S}}_{ub}(\mathbb{S}).$ Then, using (iv) it follows that, ${\mathfrak{S}}_{ub}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{ub}(\mathbb{S})\cup {\mathfrak{S}}_{ub}(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S})=$

${\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{uf}(\mathbb{S})\cup {\mathfrak{S}}_{uf}(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S})={\mathfrak{S}}_{uf}(\mathbb{T}\otimes \mathbb{S}).$Therefore, $\mathbb{T}\otimes \mathbb{S}$ satisfies property $(bz).$ ￭

By Theorems 3.7 and 6.1, we get the next corollary.

Corollary 6.2.

Let $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}) \mathit{and} \mathbb{S}\in {\mathbb{L}}_{bz}(\mathbb{Y})$, if:

1. $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T}\otimes \mathbb{S}))$, then $f(\mathbb{T}\otimes \mathbb{S})$ satisfies property $bz$;

2. $f\in \mathcal{H}(\mathfrak{S}(\mathbb{T})),g\in \mathcal{H}(\mathfrak{S}(\mathbb{S}))$, then $f(\mathbb{T})\otimes g(\mathbb{S})$ satisfies property $(bz).$

Next, we see how the property $(W_{{\Pi }_{00}^a})$ is transmitted from two tensor factors, which are $a$-isoloid operators, to their tensor product.

Theorem 6.3.

Let $\mathbb{T}\in \mathbb{L}\left(\mathbb{X}\right) \mathit{and}\mathit{ }\mathbb{S}\in \mathbb{L}\left(\mathbb{Y}\right)$ be two $a$-isoloid operators. If $\mathbb{T} \mathit{and} \mathbb{S}$ verify property $(W_{{\Pi }_{00}^a})$, then $\mathbb{T}\otimes \mathbb{S}$ verifies property $(W_{{\Pi }_{00}^a}).$

Proof.

By hypothesis result that $\mathbb{T}$ and $\mathbb{S}$ verify property $(bz),$ and so verify a-Browder’s theorem. Also, by Theorem 4.1 verify $a$-Weyl’s theorem.

On another hand, by Theorem 6.1, we get that $\mathbb{T}\otimes \mathbb{S}$ verify property $(bz),$ for that, $${\mathfrak{S}}_{uf}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_{uw}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_{ub}(\mathbb{T}\otimes \mathbb{S}).$$

Therefore, $\mathbb{T}\otimes \mathbb{S} \mathit{verifies} a$-Browder’s theorem, then by (v), we obtain that $${\mathfrak{S}}_{uw}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}){\mathfrak{S}}_{uw}(\mathbb{S})\cup {\mathfrak{S}}_{uw}(\mathbb{T}){\mathfrak{S}}_a(\mathbb{S}).$$

This, together with the fact that $\mathbb{T} \mathit{and} \mathbb{S}$ are $a$-isoloid verifying the $a$-Weyl’s theorem, it turns out that for (vi), $\mathbb{T}\otimes \mathbb{S} \mathit{verifies} a$-Weyl’s theorem.

Therefore, by Theorem 4.1, $\mathbb{T}\otimes \mathbb{S}$ verifies property $(W_{{\Pi }_{00}^a}).$ ￭

Note that if $\mathbb{T}\in \mathbb{L}(\mathbb{X})$ is finite $a$-isoloid, then ${\Pi }_{00}^a(\mathbb{T})={\Pi }_0^a(\mathbb{T}),$ this together with the fact that $0$ is an eigenvalue of the tensor product, allows to transmit the property $(g{W}_{{\Pi }_{00}^a})$ of two factors to its tensor product.

Theorem 6.4.

Let $\mathbb{T}\in \mathbb{L}(\mathbb{X}) \mathit{and} \mathbb{S}\in \mathbb{L}(\mathbb{Y})$ be two finitely $a$-isoloid operators such that $0\notin {\mathfrak{S}}_p(\mathbb{T}\otimes \mathbb{S})$. If $\mathbb{T}$ and $\mathbb{S}$ verify property $(g{W}_{{\Pi }_{00}^a})$, then $\mathbb{T}\otimes \mathbb{S}$ satisfies property $(g{W}_{{\Pi }_{00}^a}).$

Proof.

To prove that $\mathbb{T}\otimes \mathbb{S}$ satisfies property $(g{W}_{{\Pi }_{00}^a})$ it suffices to show, by [(Ben Ouidren & Zariouh, 2022), Corollary 2.9], that $\mathbb{T}\otimes \mathbb{S}$ satisfies property $(bz)$ and ${\mathcal{P}}_0^a(\mathbb{T}\otimes \mathbb{S})={\Pi }_0^a(\mathbb{T}\otimes \mathbb{S}).$ From the Theorem 6.1, $\mathbb{T}\otimes \mathbb{S}$ satisfies property $(bz).$

Note that $\mathbb{T}\otimes \mathbb{S}$ is $a$-isoloid, thus ${\mathcal{P}}_0^a(\mathbb{T}\otimes \mathbb{S})\subseteq {\Pi }_0^a(\mathbb{T}\otimes \mathbb{S}).$ On the other hand, as $0\notin {\mathfrak{S}}_p(\mathbb{T}\otimes \mathbb{S})$ so by (i), for $⋋\in \text{iso}({\mathfrak{S}}_a(\mathbb{T}\otimes \mathbb{S})),$ is $0\ne ⋋=\mu \nu ,$ for some $\mu \in \text{iso}({\mathfrak{S}}_a(\mathbb{T})) \mathit{and} \nu \in \text{iso}({\mathfrak{S}}_a(\mathbb{S})),$ furthermore $\mu \in {\Pi }_{00}^a(\mathbb{T}) \mathit{and} \nu \in {\Pi }_0^a(S).$ From $\mathbb{T}$ and $\mathbb{S}$ verify property $(g{W}_{{\Pi }_{00}^a})$ and are finitely $a$-isoloid, so; ${\Pi }_{00}^a(\mathbb{T})={\Pi }_0^a(\mathbb{T})={\mathcal{P}}_{00}^a(\mathbb{T}), {\Pi }_{00}^a(\mathbb{S})={\Pi }_0^a(\mathbb{S})={\mathcal{P}}_{00}^a(\mathbb{S}).$

Now, we consider $⋋\in {\Pi }_0^a(\mathbb{T}\otimes \mathbb{S}),$ so $⋋=\mu \nu ,$ with $\mu \in {\Pi }_{00}^a(\mathbb{T}) \mathit{and} \nu \in {\Pi }_0^a(\mathbb{S}),$ whereby (by (v)) $⋋\in {\mathcal{P}}_{00}^a(\mathbb{T}\otimes \mathbb{S}).$ Thus, ${\Pi }_0^a(\mathbb{T}\otimes \mathbb{S})\subseteq {\mathcal{P}}_0^a(\mathbb{T}\otimes \mathbb{S}).$ Therefore, $\mathbb{T}\otimes \mathbb{S}$ verifies property $(g{W}_{{\Pi }_{00}^a}).$ ￭

Through the property $(bz),$ the property $(g{W}_{{\Pi }_{00}^a})$ is transmitted to the tensor product from two factors, these factors being two $a$-polaroid operators.

Theorem 6.5.

Let $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}) \mathit{and} \mathbb{S}\in {\mathbb{L}}_{bz}(\mathbb{Y})$ be two $a$-polaroid operators, then $\mathbb{T}\otimes \mathbb{S}$ verifies property $(g{W}_{{\Pi }_{00}^a}).$

Proof.

By Theorem 6.1, we obtain that $\mathbb{T}\otimes \mathbb{S}$ verifies property $(bz).$ Since $\mathbb{T} \mathit{and} \mathbb{S}$ are $a$-polaroid operators, by [(Rashid, 2018), Lemma 2.6], we have that $\mathbb{T}\otimes \mathbb{S}$ is $a$-polaroid. Therefore, by Theorem 4.8, we get that $\mathbb{T}\otimes \mathbb{S}$ verifies property $(g{W}_{{\Pi }_{00}^a}).$￭

The above result is generalized by means of the Riesz calculation.

Theorem 6.6.

Let $\mathbb{T}\in {\mathbb{L}}_{bz}(\mathbb{X}) \mathit{and} \mathbb{S}\in {\mathbb{L}}_{bz}(\mathbb{Y})$ be two polaroid operators, such that $\mathfrak{S}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T})$ and $\mathfrak{S}(\mathbb{S})={\mathfrak{S}}_a(\mathbb{S})$, then:

1. $f(\mathbb{T}\otimes \mathbb{S})$ satisfies property $(g{W}_{{\Pi }_{00}^a})$; where $f\in {\mathcal{H}}_{nc}(\mathfrak{S}(\mathbb{T}\otimes \mathbb{S})).$

2. $f(\mathbb{T})\otimes g(\mathbb{S})$ satisfies property $(g{W}_{{\Pi }_{00}^a})$; where $f\in {\mathcal{H}}_{nc}(\mathfrak{S}(\mathbb{T}))$, and $g\in {\mathcal{H}}_{nc}(\mathfrak{S}(\mathbb{S})).$

Proof.

(i) By [(Aiena & Villafañe, 2005), Theorem 3] we have that $\mathbb{T}\otimes \mathbb{S}$ is polaroid, whereby by [(Aiena et al., 2010), Lemma 3.11] it turns out that $f(\mathbb{T}\otimes \mathbb{S})$ is polaroid. Note that $\mathfrak{S}(\mathbb{T}\otimes \mathbb{S})={\mathfrak{S}}_a(\mathbb{T}\otimes \mathbb{S})$ and so $\mathfrak{S}(f(\mathbb{T}\otimes \mathbb{S}))={\mathfrak{S}}_a(f(\mathbb{T}\otimes \mathbb{S})).$

Also, by Theorem 6.1, we get that $\mathbb{T}\otimes \mathbb{S}$ verifies property $(bz).$ Therefore, by Theorem 3.7, we obtain that $f(\mathbb{T}\otimes \mathbb{S})$ verifies property $(bz).$ Therefore, by Theorem 4.5 part (ii), we get that $f(\mathbb{T}\otimes \mathbb{S})$ verifies property $(g{W}_{{\Pi }_{00}^a}).$

(ii) Note that $f(\mathbb{T}) \mathit{and} g(\mathbb{S})$ are polaroid operators and so $a$-polaroid, since $\mathfrak{S}(f(\mathbb{T}))={\mathfrak{S}}_a(f(\mathbb{T}))$ and $\mathfrak{S}(g(\mathbb{T}))={\mathfrak{S}}_a(g(\mathbb{T})).$ On another hand, by Theorem 3.7 we get that $f(\mathbb{T})$ and $g(\mathbb{S})$ verify property $(bz).$ So, by Theorem 6.5, we conclude that $f(\mathbb{T})\otimes g(\mathbb{S})$ verifies property $(g{W}_{{\Pi }_{00}^a}).$ ￭

Example 6.7.

Consider the group algebra $L^1(G),$ with $G$a compact abelian group, thus $L^1(G)$ is a regular and Tauberian algebra, so by Example 4.9, result that every multiplier operator $\mathbb{T}$ is a polaroid operator, verifies property $(bz) \mathit{and} \mathfrak{S}(\mathbb{T})={\mathfrak{S}}_a(\mathbb{T}).$ Particularly, this happens for two convolution operators ${\mathbb{T}}_{\mu }$ and ${\mathbb{T}}_{\nu }\in L^1\left(G\right).$ Therefore, by Theorem 6.5, we get that ${\mathbb{T}}_{\mu }\otimes {\mathbb{T}}_{\nu }$ verifies property $(g{W}_{{\Pi }_{00}^a}).$ Furthermore, the Theorems 3.5, 3.6, 3.7 and 6.6, applies for ${\mathbb{T}}_{\mu }\otimes {\mathbb{T}}_{\nu }.$ Also, by Corollary 3.8, it turns out that ${\rho }_{uw}({\mathbb{T}}_{\mu }\otimes {\mathbb{T}}_{\nu })$ is a regularity.

7. Conclusions

• The property $(bz)$ is equivalent to the set of eigenvalues with the finite nullity of the operator having an empty topological interior. See Theorem 3.5.

• The property $(bz)$ and $(g{W}_{{\Pi }_{00}^a})$ are equivalent for operators a-polaroid. See Theorem 4.5.

• The operator $\mathbb{T}$ and its restriction ${\mathbb{T}}_{\mathbb{W}}$ equivalently satisfy the property $(bz)$ if the restriction is not an operator upper semi-Fredholm, or, if the ascent $p(\mathbb{T})$ and descent $q(\mathbb{T})$ are infinite. See Theorems 5.2 and 5.3.

• The property $(bz)$ is verified for the Riesz calculation. See Theorem 3.7.

• The set of operators that verify the property $(bz)$ is closed in $\mathbb{L}(\mathbb{X}).$ See Theorem 3.11.

• The property $(bz)$ is transmitted to the tensor product of two factors if each factor satisfies the property $(bz).$ See Theorem 6.1.

Statement on the use of AI

The authors declare that they have not used Artificial Intelligence (AI) in preparing this article.

Acknowledgements

We would like to thank the reviewers for their comments and corrections, which have improved the presentation of the article. We thank ESPOL professors Francisca Flores Nicolalde and Marcos Mendoza for their encouragement and support of mathematical research.

Disclosure statement

No potential conflict of interest was reported by the author(s).