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Generalized weyl's theorem for algebraically quasiparanormal operators
Journal of Inequalities and Applications volume 2012, Article number: 89 (2012)
Abstract
Let T or T* be an algebraically quasiparanormal 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 aBrowder's theorem holds for f(S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the BWeyl spectrum of T. Moreover, we show that if T is an algebraically quasiparanormal 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.
1. Introduction
Throughout this article, we assume that is an infinite dimensional separable Hilbert space. Let $B\left(\mathcal{H}\right)$ and ${B}_{0}\left(\mathcal{H}\right)$ denote, respectively, the algebra of bounded linear operators and the ideal of compact operators acting on . If $T\in B\left(\mathcal{H}\right)$ we shall write N(T) and R(T) for the null space and range of T. Also, let α(T): = dimN(T), β(T): = dimN(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 semiFredholm if it has closed range and finite dimensional null space and is called lower semiFredholm if it has closed range and its range has finite codimension. If $T\in B\left(\mathcal{H}\right)$ is either upper or lower semiFredholm, then T is called semiFredholm, and index of a semiFredholm operator$T\in B\left(\mathcal{H}\right)$ is defined by
If both 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) semiFredholm, then T is called upper (resp. lower) semiBFredholm. Moreover, if T_{ n } is Fredholm, then T is called BFredholm. T is called semiBFredholm if it is upper or lower semiBFredholm. Let T be semiBFredholm and let d be the degree of stable iteration of T. It follows from [1, Proposition 2.1] that T_{ m } is semiFredholm and i(T_{ m } ) = i(T_{ d } ) for each m ≥ d. This enables us to define the index of semiBFredholm T as the index of semiFredholm T_{ d } . Let $\mathrm{BF}\left(\mathcal{H}\right)$ be the class of all BFredholm 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 BFredholm 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 BFredholm 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
Let $a\in \mathcal{A}$. Then the Drazin spectrum is defined by
For $T\in B\left(\mathcal{H}\right)$, the smallest nonnegative integer 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
An operator $T\in B\left(\mathcal{H}\right)$ is called BWeyl if it is BFredholm of index 0. The BFredholm spectrum σ_{ BF } (T) and BWeyl spectrum σ_{ BW } (T) of T are defined by
Now, we consider the following sets:
By definition,
is the upper semiBessential approximate point spectrum and
is the left Drazin spectrum. It is well known that
where we write acc K for the accumulation points of $K\subseteq \u2102$. If we write iso K: = K \ acc K then we let
We say that an operator 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 aWeyl'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 aBrowder's theorem holds for T (in symbols, $T\in ga\mathcal{B}$) if
$${\sigma}_{a}\left(T\right)\backslash {\sigma}_{Bea}\left(T\right)={p}_{0}^{a}\left(T\right).$$
It is known ([3]) that the following set inclusions hold:
Recently, Han and Na introduced a new operator class which contains the classes of paranormal operators and quasiclass 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 quasiparanormal operators using the local spectral theory
2. Generalized Weyl's theorem for algebraically quasiparanormal 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 quasiclass A operator if
$${T}^{*}{\leftT\right}^{2}T\le {T}^{*}\left{T}^{2}\rightT.$$ 
(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 quasiclass A operators [4].
Definition 2.2. An operator $T\in B\left(\mathcal{H}\right)$ is called quasiparanormal if
We say that $T\in B\left(\mathcal{H}\right)$ is an algebraically quasiparanormal operator if there exists a nonconstant complex polynomial h such that h(T) is quasiparanormal.
In general, the following implications hold:
class A ⇒ quasiclass A ⇒ quasiparanormal;
paranormal ⇒ quasiparanormal ⇒ algebraically quasiparanormal.
In [4], it was observed that there are examples which are quasiparanormal but not paranormal, as well as quasiparanormal but not quasiclass A. We give a more simple example which is quasiparanormal but not quasiclass 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 quasiparanormal 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 quasiparanormal but not quasiclass
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 ${T}^{*}\left{T}^{2}\rightT=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill \sqrt{2}I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill \sqrt{2}I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$
On the other hand, since $\left{T}^{2}\right={T}^{*}T=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 2I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$
${T}^{*}\left{T}^{2}\rightT=\left(\begin{array}{cc}\hfill I\hfill & \hfill I\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 2I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 2I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$. Hence T is not quasiclass A.
However, since
we have
for all λ > 0. Therefore T is quasiparanormal. □
The following example provides an operator which is algebraically quasiparanormal but not quasiparanormal.
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 quasiparanormal but not quasiparanormal.
Proof. Since ${T}^{*}=\left(\begin{array}{ll}I\hfill & I\hfill \\ 0\hfill & I\hfill \end{array}\right)$, we have
Therefore
Since (2λ^{2}  10λ + 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 quasiparanormal. On the other hand, consider the complex polynomial h(z) = (z  1)^{2}. Then h(T) = 0, and hence T is algebraically quasiparanormal.
□
The following facts follow from the above definition and some well known facts about quasiparanormal operators [4]:

(i)
If $T\in B\left(\mathcal{H}\right)$ is algebraically quasiparanormal, then so is Tλ for each $\lambda \in \u2102$.

(ii)
If $T\in B\left(\mathcal{H}\right)$ is algebraically quasiparanormal and is a closed Tinvariant subspace
of , then $T\mathcal{M}$ is algebraically quasiparanormal.

(iii)
If T is algebraically quasiparanormal, then T has SVEP.

(iv)
Suppose T does not have dense range. Then we have:
T is quasiparanormal $\iff T=\left(\begin{array}{cc}A& B\\ 0& 0\end{array}\right)$ 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).
In general, the following implications hold:
However, each converse is not true. Consider the following example: let $T\in B\left({\ell}_{2}\right)$ be defined by
Then T is a compact quasinilpotent operator with α(T) = 1, and so T is isoloid. However, since q(T) = ∞, T is not polaroid.
An important subspace in local spectral theory is the quasinilpotent 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).
Let $\mathcal{P}\left(\mathcal{H}\right)$ denotes the class of all operators for which there exists $p:=p\left(\lambda \right)\in \mathbb{N}$ for which
and ${\mathcal{P}}_{1}\left(\mathcal{H}\right)$ denotes the class of all operators for which there exists $p:=p\left(\lambda \right)\in \mathbb{N}$ for which
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
Therefore, we have
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 quasiparanormal operator. Then T is nilpotent.
Proof. We first assume that T is quasiparanormal. 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].
Case II: Suppose T does not have dense range. Then we can represent T as the upper triangular matrix
where $A:=T\overline{T\left(\mathcal{H}\right)}$ is an paranormal operator. Since 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 quasiparanormal operator, then it is nilpotent. Now, we suppose T is algebraically quasiparanormal. Then there exists a nonconstant polynomial p such that p(T) is quasiparanormal. 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 quasiparanormal, 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 quasiparanormal. Then $T\in {\mathcal{P}}_{1}\left(\mathcal{H}\right)$.
Proof. Suppose T is algebraically quasiparanormal. Then h(T) is a quasiparanormal 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 quasiparanormal, 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 quasiparanormal 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 quasiparanormal operator. Then $f\left(T\right)\in g\mathcal{W}$ for each f ∈ H(σ(T)).
Proof. Suppose T is algebraically quasiparanormal. We first show that $T\in g\mathcal{W}$. Suppose that λ ∈ σ(T)\σ_{ BW } (T). Then T  λ is BWeyl but not invertible. It follows from [9, Lemma 4.1] that we can represent T  λ as the direct sum
Since T is algebraically quasiparanormal, 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 Tλ has finite ascent and descent, and so λ ∈ E(T). Conversely, suppose that λ ∈ E(T). Since T is algebraically quasiparanormal, 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 BWeyl, and so λ ∈ σ(T) \ σ_{ BW } (T). Thus σ(T) \ σ_{ BW } (T) = E(T), and hence $T\in g\mathcal{W}$.
Next, we claim that 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 quasiparanormal, 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
Since T is algebraically quasiparanormal, it follows from the proof of Theorem 2.8 that it is isoloid. Hence for any f ∈ H(σ(T)) we have
Since $T\in g\mathcal{W}$, we have
which implies that $f\left(T\right)\in g\mathcal{W}$.
Now suppose that T* is algebraically quasiparanormal. 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 quasiparanormal, 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 BWeyl, 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 quasiparanormal 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 quasiparanormal. 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 quasiparanormal and S ≺ T. Then $f\left(S\right)\in ga\mathcal{B}$ for each f ∈ H(σ(S)).
Proof. Suppose T is algebraically quasiparanormal. 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 quasiparanormal 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 quasiparanormal 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 quasiparanormal. 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(FY_{ i } ) = {μ_{ i } }. Define F_{ i } : = FY_{ i } and Ti : = TY_{ i } . Then clearly, the restrictions T_{ i } and F_{ i } commute for every i = 1, 2,...,n and
Let h be a nontrivial complex polynomial such that h(F) = 0. Then h(F_{ i } ) = h(FY_{ 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
Since T_{ i } + μ_{ i } is algebraically quasiparanormal 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 quasiparanormal, ${T}_{i}+{\mu}_{i}\in {\mathcal{P}}_{1}\left({Y}_{i}\right)$ 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 quasinilpotent 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}.
Case II: Suppose that 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 quasiparanormal, 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 BWeyl. 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} .
From Cases I and II we have
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 finitedimensional, then F is algebraic.
Corollary 3.5. Suppose $T\in B\left(\mathcal{H}\right)$ is algebraically quasiparanormal and F is an operator commuting with T such that F^{n} is a finitedimensional operator for some $n\in \mathbb{N}$. Then $T+F\in g\mathcal{W}$.
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The authors would like to express their thanks to the referee for several extremely valuable suggestions concerning the article.
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An, I.J., Han, Y.M. Generalized weyl's theorem for algebraically quasiparanormal operators. J Inequal Appl 2012, 89 (2012). https://doi.org/10.1186/1029242X201289
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Keywords
 algebraically quasiparanormal operator
 generalized Weyl's theorem
 single valued extension property