# On operators satisfying an inequality

- Salah Mecheri
^{1}Email author

**2012**:244

https://doi.org/10.1186/1029-242X-2012-244

© Mecheri; licensee Springer 2012

**Received: **27 February 2012

**Accepted: **5 October 2012

**Published: **24 October 2012

## Abstract

An operator *T* is said to be *k*-quasi-∗-class *A* if ${T}^{\ast k}(|{T}^{2}|-{|{T}^{\ast}|}^{2}){T}^{k}\ge 0$, where *k* is a natural number. Let ${d}_{A,B}\in B(B(H))$ denote either the generalized derivation ${\delta}_{A,B}={L}_{A}-{R}_{B}$ or the elementary operator ${\mathrm{\Delta}}_{A,B}={L}_{A}{R}_{B}-I$, where ${L}_{A}$ and ${R}_{B}$ are the left and right multiplication operators defined on $B(H)$ by ${L}_{A}=AX$ and ${R}_{B}=XB$ respectively. This article concerns some spectral properties of *k*-quasi-∗-class *A* operators in a Hilbert space, as the property of being hereditarily polaroid. We also establish Weyl-type theorems for *T* and ${d}_{A,B}$, where *T* is a *k*-quasi-∗-class *A* operator and *A*, ${B}^{\ast}$ are also *k*-quasi-∗-class *A* operators.

**MSC:**47B47, 47A30, 47B20, 47B10.

## Keywords

*k*-quasi-∗-class

*A*∗-paranormal operatorWeyl’s spectrum

## 1 Introduction and preliminaries

Let $B(H)$ be the algebra of all bounded linear operators acting on an infinite dimensional separable complex Hilbert space *H*. As an easy extension of normal operators, hyponormal operators have been studied by many mathematicians. Although there are many unsolved interesting problems for hyponormal operators (*e.g.*, the invariant subspace problem), one of recent trends in operator theory is studying natural extensions of hyponormal operators. So, we introduce some of these non-hyponormal operators. Recall [1, 2] that $T\in B(H)$ is called hyponormal if ${T}^{\ast}T\ge T{T}^{\ast}$, and T is called paranormal (resp., ∗-paranormal) if $\parallel {T}^{2}x\parallel \ge {\parallel Tx\parallel}^{2}$ (resp. $\parallel {T}^{2}x\parallel \ge {\parallel {T}^{\ast}x\parallel}^{2}$) for all unit vector $x\in H$. Following [2] and [3], we say that $T\in B(H)$ belongs to the class *A* if $|{T}^{2}|\ge {|T|}^{2}$. Recently Jeon and Kim [3] have considered the following new class of operators: we say that an operator $T\in B(H)$ belongs to the ∗-class *A* if $|{T}^{2}|\ge {|{T}^{\ast}|}^{2}$.

*A*operators, and ∗-class

*A*operators by ℋ, $\mathcal{PN}$, $\mathcal{P}{\mathcal{N}}^{\ast}$, $\mathcal{A}$ and ${\mathcal{A}}^{\ast}$ respectively. From [1] and [2], it is well known that

*A*operators to quasi-∗-class

*A*operators. An operator $T\in B(H)$ is said to be quasi-∗-class

*A*if ${T}^{\ast}|{T}^{2}|T\ge {T}^{\ast}{|{T}^{\ast}|}^{2}T$. If we denote this class of operators by $\mathcal{Q}{\mathcal{A}}^{\ast}$, then

*A*operators and quasi-∗-class

*A*operators, the author in [5] introduced

*k*-quasi-∗-class

*A*operators. An operator

*T*is called

*k*-quasi-∗-class

*A*if

*k*is a natural number. Let $\mathcal{K}\mathcal{Q}{\mathcal{A}}^{\ast}$ be the class of

*k*-quasi-∗-class

*A*operators. Thus,

The spectral properties of quasi-class *A* and quasi-∗-class *A* operators have been investigated by many authors in the recent years (a useful survey on the spectral properties of these operators may also be found in [6]); see also [4]. In this paper we extend to *k*-quasi-∗-class *A* operators some of these results, for instance the property of being hereditarily polaroid already observed for ∗-paranormal operators and ∗-class *A* operators defined on Hilbert spaces [5].

The fine structure of the spectrum of paranormal operators for class *A* operators or ∗-paranormal operators has been studied by several authors, in particular, for these classes of operators, it has been proved that they satisfy Weyl’s theorem; see for instance [7, 8] for paranormal operators, [9] for algebraically class *A* operators, in [5] for quasi-∗-class *A* operators. In this paper we extend these results by proving that some other variants of Weyl’s theorem hold for *k*-quasi-∗-class *A* operators; for instance, the so-called property (*w*) introduced by Rakočević in [10] and studied in [11] and [12]. All Weyl-type theorems are established for *T* and for ${d}_{A,B}$; *T* is a *k*-quasi-∗-class *A* operator and *A*, ${B}^{\ast}$ are also *k*-quasi-∗-class *A* operators.

*X*be a complex Banach space. For a bounded linear operator $T\in B(X)$, let $N(T)$ denote the null space and ran

*T*denote the range of

*T*. Let $p:=p(T)$ be the

*ascent*of an operator

*T*. (

*I.e.*, the smallest non-negative integer

*p*such that $N({T}^{p})=N({T}^{p+1})$. If such integer does not exist, we put $p(T)=\mathrm{\infty}$.) Analogously, let $q:=q(T)$ be the

*descent*of an operator

*T*;

*i.e.*, the smallest non-negative integer

*q*such that $ran{T}^{q}=ran{T}^{q+1}$, and if such integer does not exist, we put $q(T)=\mathrm{\infty}$. It is well known that if $p(T)$ and $q(T)$ are both finite, then $p(T)=q(T)$ [[13], Proposition 38.3]. Moreover, $0<p(\lambda I-T)=q(\lambda I-T)<\mathrm{\infty}$ precisely when

*λ*is a pole of the resolvent of

*T*; see Proposition 50.2 of Heuser [13]. A bounded operator $T\in B(X)$ is said to be

*polaroid*if every isolated point of the spectrum is a pole of the resolvent. $T\in B(X)$ is said to be hereditarily polaroid if the restriction of

*T*to any closed invariant subspace is polaroid. Let ${\sigma}_{\mathrm{a}}(T)$ denote the classical

*approximate point spectrum*.

*T*is said to be

*a-polaroid*if every $\lambda \in iso{\sigma}_{\mathrm{a}}(T)$ is a pole of the resolvent of

*T*. Obviously,

In [14] it has been observed that if the dual ${T}^{\prime}$ has SVEP (respectively, *T* has SVEP), then two conditions for *T* of being polaroid or *a*-polaroid (respectively, for ${T}^{\prime}$) are equivalent. The following property has a relevant role in local spectral theory and Fredholm operator theory; see the recent monographs by Laursen and Neumann [15] and [16]. A bounded operator $T\in B(X)$ is said to have the *single valued extension property* (abbreviated SVEP) if, for every open subset *G* of ℂ and any analytic function $f:G\to X$ such that $(T-z)f(z)\equiv 0$ on *G*, we have $f(z)\equiv 0$ on *G*.

*T*,

see [[16], Theorem 3.8]. In the case of Hilbert space operators, the last implication is still true if we replace ${T}^{\prime}$ with the Hilbert adjoint ${T}^{\ast}$. A bounded operator $T\in B(X)$ is said to have *Bishop’s property* (*β*) if for every open subset *G* of ℂ and every sequence ${f}_{n}:G\to H$ of *H*-valued analytic functions such that $(T-z){f}_{n}(z)$ converges uniformly to 0 in norm on compact subsets of *G*, ${f}_{n}(z)$ converges uniformly to 0 in norm on compact subsets of *G*. It is known that the property (*β*) for *T* entails that *T* has SVEP; see [15] for details.

## 2 Main results

We begin by the following lemma which is the essence of this paper and it is a structure theorem of a *k*-quasi-∗-class *A* operator *T*.

**Lemma 2.1** [5]

*Let*$T\in B(H)$

*be a*

*k*-

*quasi*-∗-

*class*

*A*

*operator*,

*the range of*${T}^{k}$

*be not dense and*

*Then* ${T}_{1}$ *is a* ∗-*class* *A* *operator*, ${T}_{3}^{k}=0$ *and* $\sigma (T)=\sigma ({T}_{1})\cup \{0\}$.

As a consequence, we obtain the following corollary.

**Corollary 2.1** *Let* $T\in B(H)$ *be a* *k*-*quasi*-∗-*class* *A* *operator*. *If* ${T}_{1}$ *is invertible*, *then* *T* *is similar to a direct sum of a* ∗-*class* *A* *operator and a nilpotent operator*.

*Proof*Since by assumption $0\notin \sigma (T)$ we have $\sigma ({T}_{1})\cap \sigma ({T}_{3})=\mathrm{\varnothing}$, then there exists an operator

*S*such that ${T}_{1}S-S{T}_{3}={T}_{2}$ [17]. Hence,

□

**Corollary 2.2** *Let* *T* *be a* *k*-*quasi*-∗-*class* *A* *operator*. *If* *T* *is quasinilpotent*, *then it must be a nilpotent operator*.

*Proof*Invoking Lemma 2.1, we find $\sigma ({T}_{1})=0$. Since ${T}_{1}$ is ∗-class

*A*, we conclude that ${T}_{1}=0$ [18]. Since ${T}_{3}^{k}=0$, a computation shows that

□

**Lemma 2.2** [5]

*Let* *M* *be a closed* *T*-*invariant subspace of* *H*. *Then the restriction* ${T}_{|M}$ *of a* *k*-*quasi*-∗-*class* *A* *operator* *T* *to* *M* *is a* *k*-*quasi*-∗-*class* *A* *operator*.

**Theorem 2.1** [5]

*Let* $T\in B(H)$ *be* *k*-*quasi*-∗-*class* *A*. *Then* *T* *satisfies Bishop’s property* (*β*), *the single valued extension property and the Dunford property* (*C*).

**Lemma 2.3** *Let* $T\in B(H)$ *be an algebraically* *k*-*quasi*-∗-*class* *A* *operator*, *and* $\sigma (T)=\{{\mu}_{0}\}$, *then* $T-{\mu}_{0}$ *is nilpotent*.

*Proof*Assume $p(T)$ is

*k*-quasi-∗-class

*A*for some nonconstant polynomial $p(z)$. Since $\sigma (p(T))=p(\sigma (T))=\{p({\mu}_{0})\}$, the operator $p(T)-p({\mu}_{0})$ is nilpotent by Corollary 2.2. Let

and hence ${(T-{\mu}_{0})}^{m{k}_{0}}=0$. □

In the following theorem, we will prove that an algebraically *k*-quasi-∗-class *A* operator is polaroid.

**Theorem 2.2** *Let* *T* *be an algebraically* *k*-*quasi*-∗-*class* *A* *operator*. *Then* *T* *is polaroid*.

*Proof*If

*T*is an algebraically

*k*-quasi-∗-class

*A*operator, then $p(T)$ is a

*k*-quasi-∗-class

*A*operator for some nonconstant polynomial

*p*. Let $\mu \in iso(\sigma (T))$, and let ${E}_{\mu}$ be the Riesz idempotent associated to

*μ*defined by

*D*is a closed disk centered at

*μ*which contains no other points of the spectrum of

*T*. Then

*T*can be represented as follows:

where $\sigma ({T}_{1})=\{\mu \}$ and $\sigma ({T}_{2})=\sigma ({T}_{2})\setminus \{\mu \}$ . Since ${T}_{1}$ is algebraically *k*-quasi-∗-class *A* operator by Lemma 2.3 and $\sigma ({T}_{1})=\{\mu \}$, it follows from Lemma 2.3 that ${T}_{1}-\lambda I$ is nilpotent. Therefore, ${T}_{1}-\mu $ has finite ascent and descent. On the other hand, since ${T}_{2}-\mu I$ is invertible, it has finite ascent and descent. Therefore, $T-\mu I$ has finite ascent and descent. Therefore, *μ* is a pole of the resolvent of *T*. Now if $\mu \in iso(\sigma (T))$, then $\mu \in \pi (T)$. Thus, $iso(\sigma (T))\in \pi (T)$, where $\pi (T)$ denotes the set of poles of the resolvent of *T*. Hence, *T* is polaroid. □

Recall that an operator *T* is said to be hereditarily polaroid if every part of it is polaroid. Hence, it follows from Lemma 2.2 that a *k*-quasi-∗-class *A* operator is hereditarily polaroid

**Corollary 2.3** *A* *k*-*quasi*-∗-*class* *A* *operator is isoloid*.

## 3 Weyl-type theorems

*X*be a complex Banach space. For every $T\in B(X)$, define

Let ${p}_{00}(T):=\sigma (T)\setminus {\sigma}_{\mathrm{b}}(T)$, *i.e.*, ${p}_{00}(T)$ is the set of all poles of the resolvent of *T*.

**Definition 3.1** A bounded operator $T\in B(X)$ is said to satisfy *Weyl’s theorem*, in symbol (W), if $\sigma (T)\setminus {\sigma}_{\mathrm{w}}(T)={\pi}_{00}(T)$. *T* is said to satisfy *a-Weyl’s theorem*, in symbol (aW), if ${\sigma}_{\mathrm{a}}(T)\setminus {\sigma}_{\mathrm{uw}}(T)={\pi}_{00}^{a}(T)$. *T* is said to satisfy the property (*w*), if ${\sigma}_{\mathrm{a}}(T)\setminus {\sigma}_{\mathrm{uw}}(T)={\pi}_{00}(T)$.

Either *a*-Weyl’s theorem or the property (*w*) entails Weyl’s theorem. The property (*w*) and *a*-Weyl’s theorem are independent; see [11].

The concept of semi-Fredholm operators has been generalized by Berkani [19, 20] in the following way: for every $T\in B(X)$ and a nonnegative integer *n*, let us denote by ${T}_{[n]}$ the restriction of *T* to ${T}^{n}(X)$ viewed as a map from the space ${T}^{n}(X)$ into itself (we set ${T}_{[0]}=T$). $T\in B(X)$ is said to be *semi-B-Fredholm* (resp. *B*-*Fredholm*, *upper semi-B-Fredholm*, *lower semi-B-Fredholm*,) if for some integer $n\ge 0$, the range ${T}^{n}(X)$ is closed and ${T}_{[n]}$ is a semi-Fredholm operator (resp. Fredholm, upper semi-Fredholm, lower semi-Fredholm). In this case ${T}_{[m]}$ is a semi-Fredholm operator for all $m\ge n$ [20]. This enables one to define the index of a semi-*B*-Fredholm as $indT=ind{T}_{[n]}$. A bounded operator $T\in B(X)$ is said to be *B-Weyl* (respectively, *upper semi-B-Weyl*, *lower semi-B-Weyl*) if for some integer $n\ge 0$, ${T}^{n}(X)$ is closed and ${T}_{[n]}$ is Weyl (respectively, upper semi-Weyl, lower semi-Weyl). In an obvious way, all the classes of operators generate spectra, for instance, the *B-Weyl spectrum* ${\sigma}_{\mathrm{bw}}(T)$ and the *upper* *B-Weyl spectrum* ${\sigma}_{\mathrm{ubw}}(T)$. Analogously, a bounded operator $T\in B(X)$ is said to be *B-Browder* (respectively, *upper semi*-*B-Browder*, *lower semi-B-Browder*) if for some integer $n\ge 0$, ${T}^{n}(X)$ is closed and ${T}_{[n]}$ is Weyl (respectively, upper semi-Browder, lower semi-Browder). The *B-Browder spectrum* is denoted by ${\sigma}_{\mathrm{bb}}(T)$, the *upper semi-B-Browder spectrum* by ${\sigma}_{\mathrm{ubb}}(T)$.

The generalized versions of Weyl-type theorems are defined as follows.

**Definition 3.2** A bounded operator $T\in B(X)$ is said to satisfy *generalized Weyl’s theorem*, in symbol, (gW), if $\sigma (T)\setminus {\sigma}_{\mathrm{bw}}(T)=E(T)$. $T\in B(X)$ is said to satisfy *generalized* *a-Weyl’s theorem*, in symbol, (gaW), if ${\sigma}_{\mathrm{a}}(T)\setminus {\sigma}_{\mathrm{ubw}}(T)={E}^{a}(T)$. $T\in L(X)$ is said to satisfy *the generalized property (w)*, in symbol, $(gw)$, if ${\sigma}_{\mathrm{a}}(T)\setminus {\sigma}_{\mathrm{ubw}}(T)=E(T)$.

*w*) and generalized

*a*-Weyl’s theorem are also independent; see [21]. Furthermore,

see [21] and [22]. The converse of all these implications in general does not hold. Furthermore, by [[23], Theorem 3.1],

(*W*) holds for *T* ⇔ Browder’s theorem holds for *T* and ${p}_{00}(T)={\pi}_{00}(T)$.

Let ${d}_{A,B}\in B(H)$ denote either the generalized derivation ${\delta}_{A,B}={L}_{A}-{R}_{B}$ or the elementary operator ${\mathrm{\Delta}}_{A,B}={L}_{A}{R}_{B}-I$, where ${L}_{A}$ and ${R}_{B}$ are the left and right multiplication operators defined on $B(H)$ by ${L}_{A}=AX$ and ${R}_{B}=XB$ respectively. We will show that if *A*, ${B}^{\ast}$ are *k*-quasi-∗-class *A*, then ${d}_{A,B}$ is polaroid and satisfies all Weyl-type theorems. For this we need the following lemmas.

**Lemma 3.1** [24]

*Let* $A,B\in B(H)$. *If* *A*, *B* *are polaroid operators*, *then* ${d}_{A,B}$ *is polaroid*.

**Lemma 3.2** *If* *A*, ${B}^{\ast}$ *are* *k*-*quasi*-∗-*class* *A* *operators*, *then* ${d}_{A,B}$ *is polaroid*.

*Proof* It is known in a Hilbert space [14] that *B* is polaroid if and only if ${B}^{\ast}$ is polaroid. Hence, it suffices to apply the previous lemma. □

Recall that an operator $T\in B(X)$ is said to have the property (*δ*) if for every open covering $(U,V)$ of ℂ, we have $X={\mathcal{H}}_{T}(\overline{U})+{\mathcal{H}}_{T}(\overline{V})$.

**Lemma 3.3** *Let* $A,B\in B(H)$. *If* *A*, *B* *have the property* (*β*), *then* ${d}_{A,B}$ *has SVEP*.

*Proof* It is known [[15], Theorem 2.5.5] that *B* satisfies the property (*β*) if and only if ${B}^{\ast}$ satisfies the property (*δ*). Since *A*, *B* have the property (*β*) by Theorem 2.1, ${B}^{\ast}$ satisfies the property (*δ*). Hence, it results from [[15], Corollary 3.6.16] that both ${L}_{A}$ and ${R}_{B}$ satisfy the Dunford property (*C*). Since ${L}_{A}$ and ${R}_{B}$ commute, hence ${L}_{A}-{R}_{B}$ and ${L}_{A}{R}_{B}$ have SVEP by [[15], Theorem 3.6.3 and Note 3.6.19]. Therefore, ${d}_{A,B}$ satisfies SVEP. □

**Corollary 3.1** *Let* $A,B\in B(H)$. *If* *A*, ${B}^{\ast}$ *are* *k*-*quasi*-∗-*class* *A* *operators*, *then* ${d}_{A,B}$ *has SVEP*.

*T*has SVEP (everywhere), the single-valued extension property, then

*T*and ${T}^{\ast}$ satisfy Browder’s (equivalently, generalized Browder’s) theorem and

*a*-Browder’s (equivalently, generalized

*a*-Browder’s) theorem. A sufficient condition for an operator

*T*satisfying Browder’s (generalized Browder’s) theorem to satisfy Weyl’s (resp. generalized Weyl’s) theorem is that

*T*is polaroid. Now since

*T*and ${T}^{\ast}$ are polaroid operators when

*T*is algebraically

*k*-quasi-∗-class

*A*, then Weyl’s theorem and generalized Weyl’s theorem hold for

*T*and ${T}^{\ast}$ when

*T*is algebraically

*k*-quasi-∗-class

*A*. Now, observe for polaroid operators

*T*satisfying generalized Weyl’s theorem,

where $\pi (T)$ is the set of poles of the resolvent of *T*. Hence, for a polaroid operator *T*, ${T}^{\ast}$ satisfies generalized Weyl’s theorem if and only if *T* satisfies generalized Weyl’s theorem if and only if *T* satisfies Weyl’s theorem if and only if ${T}^{\ast}$ satisfies Weyl’s theorem.

**Theorem 3.1**

*Let*$T,A,B\in B(H)$.

*If*

*T*,

*A*, ${B}^{\ast}$

*are algebraically*

*k*-

*quasi*-∗-

*class*

*A*,

*then the following statements are equivalent*.

- (i)
*generalized Weyl’s theorem holds for*${T}^{\ast}$ (*resp*.*for*${d}_{{A}^{\ast},{B}^{\ast}}$). - (ii)
*generalized Weyl’s theorem holds for**T*(*resp*.*for*${d}_{A,B}$). - (iii)
*Weyl’s theorem holds for**T*(*resp*.*for*${d}_{A,B}$).

Recall that a sufficient condition for an operator *T* satisfying Browder’s (generalized Browder’s) theorem to satisfy Weyl’s (resp. generalized Weyl’s) theorem is that *T* is polaroid. Observe that if $T\in B(H)$ has SVEP, then $\sigma (T)=\overline{{\sigma}_{a}({T}^{\ast})}$. Hence, if *T* has SVEP and is polaroid, then ${T}^{\ast}$ satisfies generalized *a*-Weyl’s (so, also *a*-Weyl’s) theorem [14]. It follows from Theorem 2.1 that *k*-quasi-∗-class *A* operator has SVEP. Thus, we have the following theorem.

**Theorem 3.2**

*Let*$T,A,B\in B(H)$.

- (i)
*If**T**is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then generalized**a*-*Weyl’s theorem holds for*${T}^{\ast}$ (*resp*.*for*${d}_{{A}^{\ast},{B}^{\ast}}$). - (ii)
*If*${T}^{\ast}$*is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then generalized**a*-*Weyl’s theorem holds for**T*(*resp*.*for*${d}_{{A}^{\ast},{B}^{\ast}}$).

Recall [14] that if *T* is polaroid, then *T* satisfies generalized Weyl’s theorem (resp. generalized *a*-Weyl’s theorem) if and only if *T* satisfies Weyl’s theorem (resp. *a*-Weyl’s theorem). Hence if *T* is an algebraically *k*-quasi-∗-class *A* operator, we have the following result.

**Theorem 3.3**

*Let*$T,A,B\in B(H)$.

*If*

*T*

*is algebraically*

*k*-

*quasi*-∗-

*class*

*A*

*and*

*A*, ${B}^{\ast}$

*are*

*k*-

*quasi*-∗-

*class*

*A*,

*then*

- (i)
*Weyl’s theorem holds for**T*(*resp*.*for*${d}_{A,B}$)*if and only if generalized Weyl’s theorem holds for**T*(*resp*.*for*${d}_{A,B}$). - (ii)
*a*-*Weyl’s theorem holds for*${T}^{\ast}$ (*resp*.*for*${d}_{{A}^{\ast},{B}^{\ast}}$)*if and only if generalized**a*-*Weyl’s theorem holds for*${T}^{\ast}$ (*resp*.*for*${d}_{{A}^{\ast},{B}^{\ast}}$).

**Theorem 3.4**

*Let*$T,A,B\in B(H)$.

- (i)
*If*${T}^{\ast}$*is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then Weyl’s theorem*,*a*-*Weyl’s theorem*,*generalized Weyl’s theorem and generalized**a*-*Weyl’s theorem hold for**T*(*resp*. ${d}_{A,B}$)*and these are equivalent*. - (ii)
*If**T**is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then Weyl’s theorem*,*a*-*Weyl’s theorem*,*generalized Weyl’s theorem and generalized**a*-*Weyl’s theorem hold for*${T}^{\ast}$ (*resp*. ${d}_{{A}^{\ast},{B}^{\ast}}$)*and are equivalent*.

*Proof* Since Weyl’s theorem holds for *T* (resp. for ${d}_{A,B}$). It suffices to show that Weyl’s theorem is equivalent to each one of the other Weyl-type theorems for *T* (resp. for ${d}_{A,B}$), generalized or not. Since ${T}^{\ast}$ (resp. ${d}_{{A}^{\ast},{B}^{\ast}}$) has SVEP, Weyl’s theorem and *a*-Weyl’s theorem hold for *T* (resp. for ${d}_{A,B}$) and are equivalent by [[8], Theorem 2.16]. Theorem 3.3(i) implies that Weyl’s theorem and generalized Weyl’s theorem hold for *T* (resp. for ${d}_{A,B}$) and are equivalent. Now *a*-Weyl’s theorem and generalized *a*-Weyl’s theorem hold for *T* (resp. for ${d}_{A,B}$) and are equivalent by Theorem 3.3(ii). □

Let $f\in Hol(\sigma (T))$, where $Hol(\sigma (T))$ is the space of all functions that are analytic in an open neighborhoods of $\sigma (T)$. If *T* is polaroid, then $f(T)$ is polaroid too [14]. Thus, we have

**Theorem 3.5**

*Let*$T,A,B\in B(H)$.

- (i)
*If*${T}^{\ast}$*is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then*$f({T}^{\ast})$ (*resp*. $f({d}_{{A}^{\ast},{B}^{\ast}})$)*satisfies Weyl’s theorem*,*a*-*Weyl’s theorem*,*generalized Weyl’s theorem and generalized**a*-*Weyl’s theorem*. - (ii)
*If**T**is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then*$f(T)$ (*resp*. $f({d}_{A,B})$)*satisfies Weyl’s theorem*,*a*-*Weyl’s theorem*,*generalized Weyl’s theorem and generalized**a*-*Weyl’s theorem*.

*Proof*

- (i)
If ${T}^{\ast}$ is algebraically

*k*-quasi-∗-class*A*and*A*, ${B}^{\ast}$ are*k*-quasi-∗-class*A*, then ${T}^{\ast}$ (resp. ${d}_{{A}^{\ast},{B}^{\ast}}$) is polaroid [14]. Since ${T}^{\ast}$ (resp. ${d}_{{A}^{\ast},{B}^{\ast}}$) is polaroid, the result holds by [[14], Theorem 3.12] - (ii)
If

*T*is algebraically*k*-quasi-∗-class*A*and*A*, ${B}^{\ast}$ are*k*-quasi-∗-class*A*, then $f(T)$ (resp. $f({d}_{A,B})$) is polaroid. Since*T*(resp. $f({d}_{A,B})$) is polaroid, the result holds by [[14], Theorem 3.12].

□

According to [[14], Theorem 3.12] Theorem 3.4 may be extended as follows.

**Theorem 3.6**

*Let*$T,A,B\in B(H)$.

- (i)
*If*${T}^{\ast}$*is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then Weyl’s theorem*,*a*-*Weyl’s theorem*,*generalized Weyl’s theorem and generalized**a*-*Weyl’s theorem hold for*$f(T)$ (*resp*. $f({d}_{A,B})$)*and these are equivalent*. - (ii)
*If**T**is algebraically**k*-*quasi*-∗-*class**A**and**A*, ${B}^{\ast}$*are**k*-*quasi*-∗-*class**A*,*then Weyl’s theorem*,*a*-*Weyl’s theorem*,*generalized Weyl’s theorem and generalized**a*-*Weyl’s theorem hold for*$f({T}^{\ast})$ (*resp*. $f({d}_{{A}^{\ast},{B}^{\ast}})$)*and these are equivalent*.

**Remark 3.1** According to [14], the previous results on a Weyl-type theorem still true for the property (*w*).

## Declarations

### Acknowledgements

The author would like to thank the referee for his good reading of the paper and his comments. This paper is supported by Taibah University Research Center Project (1433-808).

## Authors’ Affiliations

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