# Generalized weyl's theorem for algebraically quasi-paranormal operators

- Il Ju An
^{1}and - Young Min Han
^{1}Email author

**2012**:89

https://doi.org/10.1186/1029-242X-2012-89

© An and Han; licensee Springer. 2012

**Received: **1 December 2011

**Accepted: **17 April 2012

**Published: **17 April 2012

## Abstract

Let *T* or *T** be an algebraically quasi-paranormal operator acting on a Hilbert space. We prove: (i) generalized Weyl's theorem holds for *f*(*T*) for every *f* ∈ *H*(*σ* (*T*)); (ii) generalized *a*-Browder's theorem holds for *f*(*S*) for every *S* ≺ *T* and *f* ∈ *H*(*σ*(*S*)); (iii) the spectral mapping theorem holds for the *B*-Weyl spectrum of *T*. Moreover, we show that if *T* is an algebraically quasi-paranormal operator, then *T + F* satisfies generalized Weyl's theorem for every algebraic operator *F* which commutes with *T*.

**Mathematics Subject Classification (2010):** Primary 47A10, 47A53; Secondary 47B20.

### Keywords

algebraically quasi-paranormal operator generalized Weyl's theorem single valued extension property## 1. Introduction

*N*(

*T*) and

*R*(

*T*) for the null space and range of

*T*. Also, let

*α*(

*T*): = dim

*N*(

*T*),

*β*(

*T*): = dim

*N(T**), and let

*σ*(

*T*),

*σ*

_{ a }(

*T*),

*σ*

_{ p }(

*T*),

*π*(

*T*),

*E*(

*T*) denote the spectrum, approximate point spectrum, point spectrum of

*T*, the set of poles of the resolvent of

*T*, the set of all eigenvalues of

*T*which are isolated in

*σ*(

*T*), respectively. An operator $T\in B\left(\mathcal{H}\right)$ is called

*upper semi-Fredholm*if it has closed range and finite dimensional null space and is called

*lower semi-Fredholm*if it has closed range and its range has finite co-dimension. If $T\in B\left(\mathcal{H}\right)$ is either upper or lower semi-Fredholm, then

*T*is called

*semi-Fredholm*, and

*index of a semi-Fredholm operator*$T\in B\left(\mathcal{H}\right)$ is defined by

*α*(

*T*) and

*β*(

*T*) are finite, then

*T*is called

*Fredholm*. $T\in B\left(\mathcal{H}\right)$ is called

*Weyl*if it is Fredholm of index zero. For $T\in B\left(\mathcal{H}\right)$ and a nonnegative integer

*n*define

*T*

_{ n }to be the restriction of

*T*to

*R*(

*T*

^{ n }) viewed as a map from

*R*(

*T*

^{ n }) into

*R*(

*T*

^{ n }) (in particular

*T*

_{0}=

*T*). If for some integer

*n*the range

*R*(

*T*

^{ n }) is closed and

*T*

_{ n }is upper (resp. lower) semi-Fredholm, then

*T*is called

*upper*(resp.

*lower*)

*semi-B-Fredholm*. Moreover, if

*T*

_{ n }is Fredholm, then

*T*is called

*B-Fredholm. T*is called

*semi-B-Fredholm*if it is upper or lower semi-

*B*-Fredholm. Let

*T*be semi-

*B*-Fredholm and let

*d*be the degree of stable iteration of

*T*. It follows from [1, Proposition 2.1] that

*T*

_{ m }is semi-Fredholm and

*i*(

*T*

_{ m }) =

*i*(

*T*

_{ d }) for each

*m ≥ d*. This enables us to define the

*index of semi-B-Fredholm T*as the index of semi-Fredholm

*T*

_{ d }. Let $\mathrm{BF}\left(\mathcal{H}\right)$ be the class of all

*B*-Fredholm operators. In [2], they studied this class of operators and they proved [2, Theorem 2.7] that an operator $T\in B\left(\mathcal{H}\right)$ is

*B*-Fredholm if and only if

*T*=

*T*

_{1}⊕

*T*

_{2}, where

*T*

_{1}is Fredholm and

*T*

_{2}is nilpotent. It appears that the concept of Drazin invertibility plays an important role for the class of

*B*-Fredholm operators. Let be a unital algebra. We say that an element $x\in \mathcal{A}$ is

*Drazin invertible of degree k*if there exists an element $a\in \mathcal{A}$ such that

*Drazin spectrum*is defined by

*p*such that

*N*(

*T*

^{ p }) =

*N(Tp+*

^{1}) is called the

*ascent*of

*T*and denoted by

*p*(

*T*). If no such integer exists, we set

*p*(

*T*) =

*∞*. The smallest nonnegative integer

*q*such that

*R(T*

^{ q }) =

*R*(

*T*

^{ q }

^{+1}) is called the

*descent*of

*T*and denoted by

*q*(

*T*). If no such integer exists, we set

*q*(

*T*) =

*∞*. It is well known that

*T*is Drazin invertible if and only if it has finite ascent and descent, which is also equivalent to the fact that

*B-Weyl*if it is

*B*-Fredholm of index 0. The

*B-Fredholm spectrum σ*

_{ BF }(

*T*) and

*B-Weyl spectrum σ*

_{ BW }(

*T*) of

*T*are defined by

*B*-essential approximate point spectrum and

*K*for the accumulation points of $K\subseteq \u2102$. If we write iso

*K: = K \ acc K*then we let

*T*has the

*single valued extension property at*λ (abbreviated SVEP at λ) if for every open set

*U*containing λ the only analytic function $f:U\to \mathcal{H}$ which satisfies the equation

is the constant function *f* ≡ 0 on *U. T* has SVEP if *T* has SVEP at every point $\lambda \in \u2102$.

**Definition 1.1**. Let $T\in B\left(\mathcal{H}\right)$.

- (1)
*Generalized Weyl's theorem holds for T*(in symbols, $T\in g\mathcal{W}$) if$\sigma \left(T\right)\backslash {\sigma}_{BW}\left(T\right)=E\left(T\right).$ - (2)
*Generalized Browder's theorem holds for T*(in symbols, $T\in g\mathcal{B}$) if$\sigma \left(T\right)\backslash {\sigma}_{BW}\left(T\right)=\pi \left(T\right).$ - (3)
*Generalized a-Weyl's theorem holds for T*(in symbols, $T\in ga\mathcal{W}$) if${\sigma}_{a}\left(T\right)\backslash {\sigma}_{Bea}\left(T\right)={\pi}_{0}^{a}\left(T\right).$ - (4)
*Generalized a-Browder's theorem holds for T*(in symbols, $T\in ga\mathcal{B}$) if${\sigma}_{a}\left(T\right)\backslash {\sigma}_{Bea}\left(T\right)={p}_{0}^{a}\left(T\right).$

Recently, Han and Na introduced a new operator class which contains the classes of paranormal operators and quasi-class *A* operators [4]. In [5], it was shown that generalized Weyl's theorem holds for algebraically paranormal operators. In this article, we extend this result to algebraically quasi-paranormal operators using the local spectral theory

## 2. Generalized Weyl's theorem for algebraically quasi-paranormal operators

**Definition 2.1**. (1) An operator $T\in B\left(\mathcal{H}\right)$ is said to be

*class A*if

- (2)
*T*is called a*quasi-class A*operator if${T}^{*}{\left|T\right|}^{2}T\le {T}^{*}\left|{T}^{2}\right|T.$ - (3)An operator $T\in B\left(\mathcal{H}\right)$ is said to be
*paranormal*if${\u2225Tx\u2225}^{2}\le \u2225{T}^{2}x\u2225\u2225x\u2225\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}x\in \mathcal{H}.$

Recently, we introduced a new operator class which is a common generalization of paranormal operators and quasi-class *A* operators [4].

**Definition 2.2**. An operator $T\in B\left(\mathcal{H}\right)$ is called quasi-paranormal if

We say that $T\in B\left(\mathcal{H}\right)$ is an *algebraically quasi-paranormal* operator if there exists a non-constant complex polynomial *h* such that *h*(*T*) is quasi-paranormal.

In general, the following implications hold:

class *A* ⇒ quasi-class *A* ⇒ quasi-paranormal;

paranormal ⇒ quasi-paranormal ⇒ algebraically quasi-paranormal.

In [4], it was observed that there are examples which are quasi-paranormal but not paranormal, as well as quasi-paranormal but not quasi-class *A*. We give a more simple example which is quasi-paranormal but not quasi-class *A*. To construct this example we recall the following lemma in [4].

**Lemma 2.3**. An operator $T\in B\left(\mathcal{H}\right)$ is quasi-paranormal if and only if

**Example 2.4**. $T=\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)\in B\left({\ell}_{2}\oplus {\ell}_{2}\right)$. Then it is quasi-paranormal but not quasi-class

*A*.

*Proof*. Since ${T}^{*}=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\left|{T}^{2}\right|=\sqrt{{\left({T}^{*}\right)}^{2}{T}^{2}}=\sqrt{{\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}^{2}{\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)}^{2}=\left(\begin{array}{cc}\hfill \sqrt{2}I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}$

Therefore ${T}^{*}\left|{T}^{2}\right|T=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill \sqrt{2}I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill \sqrt{2}I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$

On the other hand, since $\left|{T}^{2}\right|={T}^{*}T=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 2I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$

${T}^{*}\left|{T}^{2}\right|T=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 2I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 2I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$. Hence *T* is not quasi-class *A*.

for all λ *>* 0. Therefore *T* is quasi-paranormal. □

The following example provides an operator which is algebraically quasi-paranormal but not quasi-paranormal.

**Example 2.5** Let $T=\left(\begin{array}{ll}I\hfill & 0\hfill \\ I\hfill & I\hfill \end{array}\right)\in B({\ell}_{2}\oplus {\ell}_{2})$. Then it is algebraically quasi-paranormal but not quasi-paranormal.

*Proof*. Since ${T}^{*}=\left(\begin{array}{ll}I\hfill & I\hfill \\ 0\hfill & I\hfill \end{array}\right)$, we have

Since (2*λ*^{2} - 10*λ* + 10)*I* is not a positive operator for *λ* = 2, ${T}^{*}\left({T}^{2*}{T}^{2}-2\lambda {T}^{*}T+{\lambda}^{2}\right)T\ngeqq 0$ for *λ >* 0. Therefore *T* is not quasi-paranormal. On the other hand, consider the complex polynomial *h*(*z*) = (*z* - 1)^{2}. Then *h*(*T*) = 0, and hence *T* is algebraically quasi-paranormal.

**□**

- (iii)
If

*T*is algebraically quasi-paranormal, then*T*has SVEP. - (iv)
Suppose

*T*does not have dense range. Then we have:

*T* is quasi-paranormal $\iff T=\left(\begin{array}{cc}A& B\\ 0& 0\end{array}\right)$ on $\mathcal{H}=\overline{T\left(\mathcal{H}\right)}\oplus N\left({T}^{*}\right)$,

where $A=T|\overline{T\left(\mathcal{H}\right)}$ is paranormal.

An operator $T\in B\left(\mathcal{H}\right)$ is called *isoloid* if iso σ(*T*) ⊆ *σ*_{
p
} (*T*) and an operator $T\in B\left(\mathcal{H}\right)$ is called *polaroid* if iso σ(*T*) ⊆ *π*(*T*).

Then *T* is a compact quasinilpotent operator with α(*T*) = 1, and so *T* is isoloid. However, since *q*(*T*) = ∞, *T* is not polaroid.

*quasi-nilpotent part*of

*T*defined by

If $T\in B\left(\mathcal{H}\right)$, then the *analytic core K*(*T*) is the set of all $x\in \mathcal{H}$ such that there exists a constant *c >* 0 and a sequence of elements ${x}_{n}\in \mathcal{H}$ such that *x*_{0} = *x*, *Tx*_{
n
} *= x*_{
n
}_{-1}, and *║x*_{
n
}*║≤ c*^{
n
}*║x║* for all $n\in \mathbb{N}$, see [6] for information on *K*(*T*).

Evidently, $\mathcal{P}\left(\mathcal{H}\right)\subseteq {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. Now we give a characterization of ${\mathcal{P}}_{1}\left(\mathcal{H}\right)$.

**Theorem 2.6**. $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$ if and only if *π*(*T*) = *E*(*T*).

*Proof*. Suppose $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$ and let λ ∈

*E*(

*T*). Then there exists $p\in \mathbb{N}$ such that

*H*

_{0}(

*T-*λ) =

*N*(

*T*- λ)

^{ p }. Since λ is an isolated point of

*σ*(

*T*), it follows from [6, Theorem 3.74] that

and hence $\mathcal{H}=N{\left(T-\lambda \right)}^{p}\oplus {\left(T-\lambda \right)}^{p}\left(\mathcal{H}\right)$, which implies, by [6, Theorem 3.6], that *p*(*T* - λ) = *q*(*T* - λ) *≤ p*. But *α*(*T* - λ) *>* 0, hence λ **∈** *π*(*T*). Therefore *E*(*T*) ⊆ *π*(*T*). Since the opposite inclusion holds for every operator *T*, we then conclude that *π*(*T*) = *E*(*T*). Conversely, suppose *π*(*T*) = *E*(*T*). Let λ **∈** *E*(*T*). Then *p* : = *p*(*T* - λ) = *q*(*T* - λ) *<* ∞. By [6, Theorem 3.74], *H*_{0}(*T* - λ) = *N*(*T* - λ)^{
p
}. Therefore $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. □

From Theorem 2.6, we can give a simple example which belongs to ${\mathcal{P}}_{1}\left(\mathcal{H}\right)$ but not $\mathcal{P}\left(\mathcal{H}\right)$. Let *U* be the unilateral shift on ${\ell}_{2}$ and let *T = U**. Then *T* does not have SVEP at 0, and so *H*_{0}(*T*) is not closed. Therefore $T\notin \mathcal{P}\left(\mathcal{H}\right)$. However, since $\sigma \left(T\right)=\stackrel{\u0304}{D},\pi \left(T\right)=E\left(T\right)=\mathrm{0\u0338}$, where
is an open unit disk in
. Hence $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$ by Theorem 2.6.

Before we state our main theorem (Theorem 2.9) in this section, we need some preliminary results.

**Lemma 2.7**. Let $T\in B\left(\mathcal{H}\right)$ be a quasinilpotent algebraically quasi-paranormal operator. Then *T* is nilpotent.

*Proof*. We first assume that *T* is quasi-paranormal. We consider two cases:

Case I: Suppose *T* has dense range. Then clearly, it is paranormal. Therefore *T* is nilpotent by [7, Lemma 2.2].

*T*does not have dense range. Then we can represent

*T*as the upper triangular matrix

*T*is quasinilpotent,

*σ*(

*T*) = {0}. But

*σ*(

*T*) =

*σ*(

*A*) ∪ {0}, hence

*σ*(

*A*) = {0}. Since

*A*is paranormal,

*A*= 0 and therefore

*T*is nilpotent. Thus if

*T*is a quasinilpotent quasi-paranormal operator, then it is nilpotent. Now, we suppose

*T*is algebraically quasi-paranormal. Then there exists a nonconstant polynomial

*p*such that

*p*(

*T*) is quasi-paranormal. If

*p*(

*T*) has dense range, then

*p*(

*T*) is paranormal. So

*T*is algebraically paranormal, and hence

*T*is nilpotent by [7, Lemma 2.2]. If

*p*(

*T*) does not have dense range, we can represent

*p*(

*T*) as the upper triangular matrix

where $C:=p\left(T\right)|\overline{p\left(T\right)\left(\mathcal{H}\right)}$ is paranormal. Since *T* is quasinilpotent, *σ*(*p*(*T*)) = *p*(*σ*(*T*)) = {*p*(0)}. But *σ*(*p*(*T*)) = *σ*(*C*)∪{0} by [8, Corollary 8], hence *σ*(*C*)∪{0} = {*p*(0)}. So *p*(0) = 0, and hence *p*(*T*) is quasinilpotent. Since *p*(*T*) is quasi-paranormal, by the previous argument *p*(*T*) is nilpotent. On the other hand, since *p*(0) = 0, *p*(*z*) = *cz*^{
m
} (*z* - λ_{1})(*z* - λ_{2}) ... (*z* - λ_{
n
}) for some natural number *m*. Therefore *p*(*T*) = *cT*^{
m
} (*T* - λ1)(*T* - λ_{2}) ... (*T* - λ_{
n
}). Since *p*(*T*) is nilpotent and *T* - λ_{
i
}is invertible for every λ_{
i
}≠ 0, *T* is nilpotent. This completes the proof. □

**Theorem 2.8**. Let $T\in B\left(\mathcal{H}\right)$ be algebraically quasi-paranormal. Then $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$.

*Proof*. Suppose

*T*is algebraically quasi-paranormal. Then

*h*(

*T*) is a quasi-paranormal operator for some nonconstant complex polynomial

*h*. Let λ ∈

*E*(

*T*). Then

*λ*is an isolated point of

*σ*(

*T*) and

*α*(

*T -*λ)

*>*0. Using the spectral projection $P:=\frac{1}{2\pi i}{\int}_{\partial D}{\left(\mu -T\right)}^{-1}d\mu $, where

*D*is a closed disk of center λ which contains no other points of

*σ*(

*T*), we can represent

*T*as the direct sum

Since *T*_{1} is algebraically quasi-paranormal, so is *T*_{1} - λ. But *σ*(*T*_{1} - λ) = {0}, it follows from Lemma 2.7 that *T*_{1} - λ is nilpotent. Therefore *T*_{1} - λ has finite ascent and descent. On the other hand, since *T*_{2} - λ is invertible, clearly it has finite ascent and descent. Therefore λ is a pole of the resolvent of *T*, and hence λ ∈ *π*(*T*). Hence *E*(*T*) ⊆ *π*(*T*). Since *π*(*T*) ⊆ *E*(*T*) holds for any operator *T*, we have *π*(*T*) = *E*(*T*). It follows from Theorem 2.6 that $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. **□**

We now show that generalized Weyl's theorem holds for algebraically quasi-paranormal operators. In the following theorem, recall that *H*(*σ*(*T*)) is the space of functions analytic in an open neighborhood of *σ*(*T*).

**Theorem 2.9**. Suppose that *T* or *T** is an algebraically quasi-paranormal operator. Then $f\left(T\right)\in g\mathcal{W}$ for each *f* ∈ *H*(*σ*(*T*)).

*Proof*. Suppose

*T*is algebraically quasi-paranormal. We first show that $T\in g\mathcal{W}$. Suppose that λ ∈

*σ*(

*T*)\

*σ*

_{ BW }(

*T*). Then

*T*- λ is

*B*-Weyl but not invertible. It follows from [9, Lemma 4.1] that we can represent

*T -*λ as the direct sum

*T*is algebraically quasi-paranormal, it has SVEP. So

*T*

_{1}and

*T*

_{2}have both finite ascent. But

*T*

_{1}is Weyl, hence

*T*

_{1}has finite descent. Therefore

*T*-λ has finite ascent and descent, and so

*λ*∈

*E*(

*T*). Conversely, suppose that

*λ*∈

*E*(

*T*). Since

*T*is algebraically quasi-paranormal, it follows from Theorem 2.8 that $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. Since

*π*(

*T*) =

*E*(

*T*) by Theorem 2.6,

*λ*∈

*E*(

*T*). Therefore

*T - λ*has finite ascent and descent, and so we can represent

*T - λ*as the direct sum

Therefore *T - λ* is *B*-Weyl, and so *λ* ∈ *σ*(*T*) *\ σ*_{
BW
} (*T*). Thus *σ*(*T*) \ *σ*_{
BW
} (*T*) = *E*(*T*), and hence $T\in g\mathcal{W}$.

*σ*

_{ BW }(

*f*(

*T*)) =

*f*(

*σ*

_{ BW }(

*T*)) for each

*f*∈

*H*(

*σ*(

*T*)). Since $T\in g\mathcal{W},\phantom{\rule{2.77695pt}{0ex}}T\in g\mathcal{B}$. It follows from [5, Theorem 2.1] that

*σ*

_{ BW }(

*T*) =

*σ*

_{ D }(

*T*). Since

*T*is algebraically quasi-paranormal,

*f*(

*T*) has SVEP for each

*f*∈

*H*(

*σ*(

*T*)). Hence $f\left(T\right)\in g\mathcal{B}$ by [5, Theorem 2.9], and so

*σ*

_{ BW }(

*f*(

*T*)) =

*σ*

_{ D }(

*f*(

*T*)). Therefore we have

*T*is algebraically quasi-paranormal, it follows from the proof of Theorem 2.8 that it is isoloid. Hence for any

*f*∈

*H*(

*σ*(

*T*)) we have

which implies that $f\left(T\right)\in g\mathcal{W}$.

*T** is algebraically quasi-paranormal. We first show that $T\in g\mathcal{W}$. Let

*λ*∈

*σ*(

*T*)

*\ σ*

_{ BW }(

*T*). Observe that $\sigma \left({T}^{*}\right)=\overline{\sigma \left(T\right)}$ and ${\sigma}_{BW}\left({T}^{*}\right)=\overline{{\sigma}_{BW}\left(T\right)}$. So $\overline{\lambda}\in \sigma \left({T}^{*}\right)\backslash {\sigma}_{BW}\left({T}^{*}\right)$, and so $\overline{\lambda}\in E\left({T}^{*}\right)$ because ${T}^{*}\in g\mathcal{W}$. Since

*T** is algebraically quasi-paranormal, it follows from Theorem 2.8 that $\overline{\lambda}\in \pi \left({T}^{*}\right)$. Hence

*T - λ*has finite ascent and descent, and so

*λ*∈

*E*(

*T*). Conversely, suppose

*λ*∈

*E*(

*T*). Then

*λ*is an isolated point of

*σ*(

*T*) and

*α*(

*T - λ*)

*>*0. Since $\sigma \left({T}^{*}\right)=\overline{\sigma \left(T\right)},\stackrel{\u0304}{\lambda}$ is an isolated point of

*σ*(

*T**). Since

*T** is isoloid, $\overline{\lambda}\in E\left({T}^{*}\right)$. But

*E*(

*T**) =

*π*(

*T**) by Theorem 2.8, hence we have

*T - λ*has finite ascent and descent. Therefore we can represent

*T - λ*as the direct sum

Therefore *T - λ* is *B*-Weyl, and so *λ* ∈ *σ*(*T*) *\ σ*_{
BW
} (*T*). Thus *σ*(*T*) *\ σ*_{
BW
} (*T*) = *E(T*), and hence $T\in g\mathcal{W}$. If *T** is algebraically quasi-paranormal then *T* is isoloid. It follows from the first part of the proof that $f\left(T\right)\in g\mathcal{W}$. This completes the proof. □

From the proof of Theorem 2.9 and [10, Theorem 3.4], we obtain the following useful consequence.

**Corollary 2.10**. Suppose

*T*or

*T**is algebraically quasi-paranormal. Then

An operator $X\in B\left(\mathcal{H}\right)$ is called a *quasiaffinity* if it has trivial kernel and dense range. $S\in B\left(\mathcal{H}\right)$ is said to be a *quasiaffine transform of* $T\in B\left(\mathcal{H}\right)$(notation: *S* ≺ *T*) if there is a quasiaffinity $X\in B\left(\mathcal{H}\right)$ such that *XS = TX*. If both *S* ≺ *T* and *T* ≺ *S*, then we say that *S* and *T* are *quasisimilar*.

**Corollary 2.11**. Suppose *T* is algebraically quasi-paranormal and *S* ≺ *T*. Then $f\left(S\right)\in ga\mathcal{B}$ for each *f* ∈ *H*(*σ*(*S*)).

*Proof*. Suppose *T* is algebraically quasi-paranormal. Then *T* has SVEP. Since *S* ≺ *T*, *f*(*S*) has SVEP by [7, Lemma 3.1]. It follows from [11, Theorem 3.3.6] that *f*(*S*) has SVEP. Therefore $f\left(S\right)\in ga\mathcal{B}$ by [12, Corollary 2.5]. □

## 3. Generalized Weyl's theorem for perturbations of algebraically quasi-paranormal operators

An operator *T* is said to be *algebraic* if there exists a nontrivial polynomial *h* such that *h*(*T*) = 0. From the spectral mapping theorem it easily follows that the spectrum of an algebraic operator is a finite set. It is known that generalized Weyl's theorem is not generally transmitted to perturbation of operators satisfying generalized Weyl's theorem. In [13], they proved that if *T* is paranormal and *F* is an algebraic operator commuting with *T*, then Weyl's theorem holds for *T + F*. We now extend this result to generalized Weyl's theorem for algebraically quasi-paranormal operators. We begin with the following lemma.

**Lemma 3.1**. Let $T\in B\left(\mathcal{H}\right)$. Then the following statements are equivalent:

- (1)
$T\in g\mathcal{W}$;

- (2)
*T*has SVEP at every $\lambda \in \u2102\backslash {\sigma}_{BW}\left(T\right)$ and*π*(*T*) =*E*(*T*).

*Proof*. Observe that $T\in g\mathcal{B}$ if and only if *σ*_{
BW
} (*T*) = *σ*_{
D
} (*T*). *So* $T\in g\mathcal{B}$ if and only if *T* has SVEP at every $\lambda \in \u2102\backslash {\sigma}_{BW}\left(T\right)$. Therefore we obtain the desired conclusion. □

From this lemma, we obtain the following corollary

**Corollary 3.2**. Let $T\in B\left(\mathcal{H}\right)$. Suppose

*T*has SVEP. Then

*Proof*. Since *T* has SVEP, $T\in g\mathcal{B}$*by* Lemma 3.1. So *σ*(*T*) *\ σ*_{
BW
} (*T*) = *π*(*T*). Therefore $T\in g\mathcal{W}$ if and only if $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$ by Theorem 2.6. □

**Lemma 3.3**. Suppose $T\in B\left(\mathcal{H}\right)$ and *N* is nilpotent such that *TN = NT*. Then $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$ if and only if $T+N\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$.

*Proof*. Suppose

*N*

^{ p }= 0 for some $p\in \mathbb{N}$. Observe that without any assumption on

*T*we have

Suppose now that $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$, or equivalently *π*(*T*) = *E(T*). We show first *E*(*T*) = *E*(*T+N*). Let *λ* ∈ *E*(*T*). Without loss of generality, we may assume that *λ* = 0. From *σ*(*T+N*) = *σ*(*T*), we see that 0 is an isolated point of *σ*(*T+N*). Since 0 ∈ *E*(*T*), *α*(*T*) *>* 0 and hence by the first inclusion in (3.3.1) we have *α*(*T+N*)^{
p
}> 0. Therefore *α*(*T+N*) *>* 0, and hence 0 ∈ *E*(*T+N*). Thus the inclusion *E*(*T*) ⊆ *E*(*T* + *N*) is proved. To show the opposite inclusion, assume that 0 ∈ *E*(*T* + *N*). Then 0 is an isolated point of *σ*(*T*) because *σ*(*T* + *N*) = *σ*(*T*). Since *α*(*T* + *N*) *>* 0, the second inclusion in (3.3.1) entails that *α*(*T*^{
p
} ) *>* 0. Therefore *α*(*T*) *>* 0, and hence 0 ∈ *E*(*T*). So the equality *E*(*T*) = *E*(*T* + *N*) is proved. Suppose $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. Then *π*(*T*) = *E*(*T*) by Theorem 2.6, and so *π*(*T* + *N*) = *π*(*T*) = *E*(*T*) = *E*(*T* + *N*). Therefore $T+N\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. Conversely, if $T+N\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$ by symmetry we have *π*(*T*) = *π*(*T* + *N*) = *E*(*T* + *N*) = *E*((*T + N*)-*N*) = *E*(*T*), so the proof is complete. □

The following theorem is a generalization of [13, Theorem 2.5]. The proof of the following theorem is strongly inspired to that of it.

**Theorem 3.4**. Suppose *T* is algebraically quasi-paranormal. If *F* is algebraic with *TF = FT*, then $T+F\in g\mathcal{W}$.

*Proof*. Since

*F*is algebraic,

*σ*(

*F*) is finite. Let

*σ*(

*F*) = {

*μ*

_{1},

*μ*

_{2},...,

*μ*

_{ n }}. Denote by

*P*

_{ i }the spectral projection associated with

*F*and the spectral set {

*μ*

_{ i }}. Let

*Y*

_{ i }: =

*R*(

*P*

_{ i }) and

*Z*

_{ i }: =

*N*(

*P*

_{ i }). Then

*H = Y*

_{ i }⊕

*Z*

_{ i }and the closed subspaces

*Y*

_{ i }and

*Z*

_{ i }are invariant under

*T*and

*F*. Moreover,

*σ*(

*F|Y*

_{ i }) = {

*μ*

_{ i }}. Define

*F*

_{ i }: =

*F|Y*

_{ i }and

*Ti*: =

*T|Y*

_{ i }. Then clearly, the restrictions

*T*

_{ i }and

*F*

_{ i }commute for every

*i*= 1, 2,...,

*n*and

*h*be a nontrivial complex polynomial such that

*h*(

*F*) = 0. Then

*h*(

*F*

_{ i }) =

*h*(

*F|Y*

_{ i }) =

*h*(

*F*)

*|Y*

_{ i }= 0, and from {0} =

*σ*(

*h*(

*F*

_{ i })) =

*h*(

*σ*(

*F*

_{ i })) =

*h*({

*μ*

_{ i }}), we obtain that

*h*(

*μ*

_{ i }) = 0. Write

*h*(

*μ*) = (

*μ - μ*

_{ i })

^{ m }

*g*(

*μ*) with

*g*(

*μ*

_{ i }) = 0. Then 0 =

*h*(

*F*

_{ i }) = (

*F - μ*

_{ i })

^{ m }

*g*(

*F*

_{ i }), where

*g*(

*F*

_{ i }) is invertible. Hence

*N*

_{ i }: =

*F*

_{ i }

*- μ*

_{ i }are nilpotent for all

*i*= 1, 2,...,

*n*. Observe that

*T*

_{ i }+

*μ*

_{ i }is algebraically quasi-paranormal for all

*i*= 1, 2,...,

*n*,

*T*

_{ i }+

*μ*

_{ i }has SVEP. Moreover, since

*N*

_{ i }is nilpotent with

*T*

_{ i }

*N*

_{ i }

*= N*

_{ i }

*T*

_{ i }, it follows from [6, Corollary 2.12] that

*T*

_{ i }

*+ N*

_{ i }

*+ μ*

_{ i }has SVEP, and hence

*T*

_{ i }+

*F*

_{ i }has SVEP. From [6, Theorem 2.9] we obtain that

Now, we show that $T+F\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. Since *T*_{
i
} *+ μ*_{
i
} is algebraically quasi-paranormal, ${T}_{i}+{\mu}_{i}\in {\mathcal{P}}_{1}\left({Y}_{i}\right)$ by Theorem 2.8. By Lemma 3.3 and (3.4.1), ${T}_{i}+{F}_{i}\in {\mathcal{P}}_{1}\left({Y}_{i}\right)$ for every *i* = 1, 2,...,*n*. Now assume that λ_{0} ∈ *E(T* + *F*). Fix $i\in \mathbb{N}$ such that 1 *≤ i ≤ n*. Since the equality *T*_{
i
} + *N*_{
i
} - λ_{0} + *μ*_{
i
} *= T*_{
i
} + *F*_{
i
} - λ_{0} holds, we consider two cases:

Case I: Suppose that *T*_{
i
} - λ_{0} + *μ*_{
i
} is invertible. Since *N*_{
i
} is quasi-nilpotent commuting with *T*_{
i
} - λ_{0} + *μ*_{
i
} , it is clear that *T*_{
i
} *+ F*_{
i
} - λ_{0} is also invertible. Hence *H*_{0}(*T*_{
i
} + *F*_{
i
} - λ_{0}) = *N(T*_{
i
} + *F*_{
i
} - λ_{0}) = {0}.

*T*

_{ i }- λ

_{0}+

*μ*

_{ i }is not invertible. Then λ

_{0}-

*μ*

_{ i }∈

*σ*(

*T*

_{ i }). We claim that λ

_{0}∈

*E*(

*T*

_{ i }+

*F*

_{ i }). Note that λ

_{0}∈

*σ*(

*T*

_{ i }+

*μ*

_{ i }) =

*σ*(

*T*

_{ i }+

*F*

_{ i }). Since

*σ*(

*T*

_{ i }+

*F*

_{ i }) ∈

*σ*(

*T*+

*F*) and λ

_{0}∈ iso

*σ*(

*T*+

*F*), λ

_{0}∈ iso

*σ(T*

_{ i }

*+ N*

_{ i }

*+ μ*

_{ i }). Therefore λ

_{0}-

*μ*

_{ i }∈ iso

*σ*(

*T*

_{ i }+

*N*

_{ i }) = iso

*σ*(

*T*

_{ i }). Since

*T*

_{ i }- λ

_{0}+

*μ*

_{ i }is algebraically quasi-paranormal, λ

_{0}-

*μ*

_{ i }∈

*π*(

*T*

_{ i }). Since

*π*(

*T*

_{ i }) =

*E*(

*T*

_{ i }) by Theorem 2.6 and ${T}_{i}\in g\mathcal{W}$ by Theorem 2.9,

*λ*

_{0}-

*μ*

_{ i }∈

*E*(

*T*

_{ i }) =

*σ*(

*T*

_{ i })

*\ σ*

_{ BW }(

*T*

_{ i }). But

*N*

_{ i }is nilpotent with

*T*

_{ i }

*N*

_{ i }

*= N*

_{ i }

*T*

_{ i }, hence

*σ*

_{ D }(

*T*

_{ i }) =

*σ*

_{ D }(

*T*

_{ i }+

*N*

_{ i }) and ${T}_{i}+{N}_{i}\in g\mathcal{B}$. Therefore we have

*σ*

_{ BW }(

*T*

_{ i }+

*N*

_{ i }) =

*σ*

_{ D }(

*T*

_{ i }+

*N*

_{ i }). Hence

Hence *T*_{
i
} + *F*_{
i
} - λ_{0} is *B-Weyl*. Assume to the contrary that *T*_{
i
} + *F*_{
i
} - λ_{0} is injective. Then *β(T*_{
i
} + *F*_{
i
} - λ_{0}) = *α(T*_{
i
} + *F*_{
i
} - λ_{0}) = 0. Therefore *T*_{
i
} + *F*_{
i
} - λ_{0} is invertible, and so λ_{0} ∉ *σ*(*T*_{
i
}*+F*_{
i
} ). This is a contradiction. Hence λ_{0} ∈ *E*(*T*_{
i
} + *F*_{
i
} ). Since ${T}_{i}+{F}_{i}\in {\mathcal{P}}_{1}\left({Y}_{i}\right)$ by Theorem 2.6, there exists a positive integer *m*_{
i
} such that *H*_{
0
}*(T*_{
i
} + *F*_{
i
} - λ_{0}) = *N(T*_{
i
} + *F*_{
i
} - λ_{0}) ^{
mi
} .

where *m* : = max{*m*_{1},*m*_{2},...,*m*_{
n
} }. Since the last equality holds for every λ_{0} ∈ *E*(*T* + *F*), $T+F\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$. Therefore $T+F\in g\mathcal{W}$*by* Corollary 3.2. □

It is well known that if for an operator $F\in B\left(\mathcal{H}\right)$ there exists a natural number *n* for which *F*^{
n
} is finite-dimensional, then *F* is algebraic.

**Corollary 3.5**. Suppose $T\in B\left(\mathcal{H}\right)$ is algebraically quasi-paranormal and *F* is an operator commuting with *T* such that *F*^{
n
} is a finite-dimensional operator for some $n\in \mathbb{N}$. Then $T+F\in g\mathcal{W}$.

## Declarations

### Acknowledgements

The authors would like to express their thanks to the referee for several extremely valuable suggestions concerning the article.

## Authors’ Affiliations

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