Generalized weyl's theorem for algebraically quasi-paranormal operators

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.

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 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

$$i\left(T\right):=\alpha \left(T\right)-\beta \left(T\right).$$

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(Tⁿ) viewed as a map from R(Tⁿ ) into R(Tⁿ ) (in particular T₀ = T). If for some integer n the range R(Tⁿ ) 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 = T₁ ⊕ T₂, where T₁ is Fredholm and T₂ 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

$$x^{k}ax=x^k,axa=a,\text{and}xa=ax.$$

Let $a\in \mathcal{A}$. Then the Drazin spectrum is defined by

$${\sigma }_D\left(a\right):=\left\{\lambda \in ℂ:a-\lambda \text{is}\text{not}\text{Drazin}\text{invertible}\right\}.$$

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

$$T=T_1\oplus T_2,\text{where}{T}_{\text{1}}\text{is}\text{invertible}\text{and}{T}_{\text{2}}\text{is}\text{nilpotent}\text{.}$$

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

$${\sigma }_{BF}\left(T\right):=\left\{\lambda \in ℂ:T-\lambda \text{is}\text{not}B-\text{Fredholm}\right\},$$

$${\sigma }_{BW}\left(T\right):=\left\{\lambda \in ℂ:T-\lambda \text{is}\text{not}B-\text{Weyl}\right\}.$$

Now, we consider the following sets:

$$\begin{array}{l}B{F}_{+}(\mathscr{H}):=\{T\in B(\mathscr{H}):T\text{is}\text{upper}\text{semi-}B\text{-Ferdholm},}\\ B{F}_{+}^{-}(\mathscr{H}):=\{T\in B(\mathscr{H}):T\in B{F}_{+}(\mathscr{H})\text{and}i(T)\le 0\text{},}\\ LD(\mathscr{H}):=\{T\in B(\mathscr{H}):p(T)<\infty \text{and}R(T^{p(T)+1})\text{is closed}}\text{.}\end{array}$$

By definition,

$${\sigma }_{Bea}\left(T\right):=\left\{\lambda \in ℂ:T-\lambda \notin B{F}_{+}^{-}\left(\mathcal{H}\right)\right\},$$

is the upper semi-B-essential approximate point spectrum and

$${\sigma }_{LD}\left(T\right):=\left\{\lambda \in ℂ:T-\lambda \notin LD\left(\mathcal{H}\right)\right\}$$

is the left Drazin spectrum. It is well known that

$${\sigma }_{Bea}\left(T\right)\subseteq {\sigma }_{LD}\left(T\right)={\sigma }_{Bea}\left(T\right)\cup \text{acc}{\sigma }_a\left(T\right)\subseteq {\sigma }_D\left(T\right),$$

where we write acc K for the accumulation points of $K\subseteq ℂ$. If we write iso K: = K \ acc K then we let

$$\begin{array}{ll}\hfill p_0^a\left(T\right)& :=\left\{\lambda \in {\sigma }_{\alpha }\left(T\right):T-\lambda \in LD\left(\mathcal{H}\right)\right\},\\ \hfill {\pi }_0^a\left(T\right)& :=\left\{\lambda \in \text{iso}{\sigma }_a\left(T\right):\lambda \in {\sigma }_p(T)\right\}.\end{array}$$

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

$$\left(T-\lambda \right)f\left(\lambda \right)=0$$

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

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

  $${\sigma }_a\left(T\right)\backslash {\sigma }_{Bea}\left(T\right)=p_0^a\left(T\right).$$

It is known ([3]) that the following set inclusions hold:

$$\begin{array}{l}ga-\text{Weyl's}\text{theorem}\Rightarrow ga-\text{Browder's}\text{theorem}\\ \Downarrow \Downarrow \\ g-\text{Weyl}\text{'}\text{s}\text{theorem}\Rightarrow g-\text{Browder}\text{'}\text{s}\text{theorem}\end{array}$$

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

$${\left|T\right|}^2\le \left|T^2\right|.$$

• (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

  $${∥Tx∥}^2\le ∥T^{2}x∥∥x∥\text{for}\text{all}x\in \mathcal{H}.$$

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

$${∥T^{2}x∥}^2\le ∥T^{3}x∥∥Tx∥\text{for}\text{all}x\in \mathcal{H}.$$

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

$$T^{*}\left({T^2}^{*}{T}^2-2\lambda T^{*}T+{\lambda }^2\right)T\ge 0\text{for}\text{all}\lambda >0.$$

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.

However, since

$$T^{2*}{T}^2-2\lambda T^{*}T+{\lambda }^2=\left(\begin{array}{cc}\hfill \left(2-4\lambda +{\lambda }^2\right)I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\lambda }^{2}I\hfill \end{array}\right),$$

we have

$$T^{*}\left(T^{2*}{T}^2-2\lambda T^{*}T+{\lambda }^2\right)T=\left(\begin{array}{cc}\hfill 2{\left(1-\lambda \right)}^{2}I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\ge 0$$

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

$$T^{2*}{T}^2-2\lambda T^{*}T+{\lambda }^2=\left(\begin{array}{cc}({\lambda }^2-4\lambda +5)I& (-2\lambda +2)I\\ (-2\lambda +2)I& ({\lambda }^2-2\lambda +1)I\end{array}\right).$$

Therefore

$$T^{*}(T^{2*}{T}^2-2\lambda T^{*}T+{\lambda }^2)T=\left(\begin{array}{cc}(2{\lambda }^2-10\lambda +10)I& ({\lambda }^2-4\lambda +3)I\\ ({\lambda }^2-4\lambda +3)I& ({\lambda }^2-2\lambda +1)I\end{array}\right).$$

Since (2λ² - 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 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.

□

The following facts follow from the above definition and some well known facts about quasi-paranormal operators [4]:

• (i) If $T\in B\left(\mathcal{H}\right)$ is algebraically quasi-paranormal, then so is T-λ for each $\lambda \in ℂ$.

• (ii) If $T\in B\left(\mathcal{H}\right)$ is algebraically quasi-paranormal and is a closed T-invariant subspace

of , then $T|\mathcal{M}$ 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).

In general, the following implications hold:

$$T\text{polaroid}\Rightarrow T\text{isoloid}\text{.}$$

However, each converse is not true. Consider the following example: let $T\in B\left({\ell }_2\right)$ be defined by

$$T\left(x_1,x_2,x_3,\dots \right)=\left(\frac{1}{2}{x}_2,\frac{1}{3}{x}_3,\dots \right).$$

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 quasi-nilpotent part of T defined by

$$H_0\left(T\right):=\left\{x\in \mathcal{H}:\underset{\text{n}\to \infty }{\text{lim}}{∥T^{n}x∥}^{\frac{1}{n}}=0\right\}.$$

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 x₀ = x, Tx _n = x _{n -1}, and ║x _n ║≤ cⁿ║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

$$H_0\left(T-\lambda \right)=N{\left(T-\lambda \right)}^p\text{for}\text{all}\lambda \in ℂ,$$

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

$$H_0\left(T-\lambda \right)=N{\left(T-\lambda \right)}^p\text{for}\text{all}\lambda \in E\left(T\right).$$

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 H₀(T- λ) = N(T - λ)^p. Since λ is an isolated point of σ(T), it follows from [6, Theorem 3.74] that

$$\mathcal{H}=H_0\left(T-\lambda \right)\oplus K\left(T-\lambda \right)=N{\left(T-\lambda \right)}^p\oplus K\left(T-\lambda \right).$$

Therefore, we have

$${\left(T-\lambda \right)}^p\left(\mathcal{H}\right)={\left(T-\lambda \right)}^p\left(K\left(T-\lambda \right)\right)=K\left(T-\lambda \right),$$

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], H₀(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 H₀(T) is not closed. Therefore $T\notin \mathcal{P}\left(\mathcal{H}\right)$. However, since $\sigma \left(T\right)=\bar{D},\pi \left(T\right)=E\left(T\right)=\mathrm{0̸}$, 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].

Case II: Suppose T does not have dense range. Then we can represent T as the upper triangular matrix

$$T=\left(\begin{array}{cc}A& B\\ 0& 0\end{array}\right)\text{on}\mathscr{H}=\overline{T\left(\mathscr{H}\right)}\oplus N\left(T^{*}\right),$$

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 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

$$p\left(T\right)=\left(\begin{array}{cc}C& D\\ 0& 0\end{array}\right)\text{on}\mathscr{H}=\overline{p\left(T\right)\left(\mathscr{H}\right)}\oplus N\left(p{\left(T\right)}^{*}\right),$$

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 - λ₁)(z - λ₂) ... (z - λ _n ) for some natural number m. Therefore p(T) = cT^m (T - λ1)(T - λ₂) ... (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

$$T=\left(\begin{array}{cc}{T}_1& 0\\ 0& T_2\end{array}\right),\text{where}\sigma \left(T_1\right)=\left\{\lambda \right\}\text{and}\sigma \left(T_2\right)=\sigma \left(T\right)\backslash \left\{\lambda \right\}.$$

Since T₁ is algebraically quasi-paranormal, so is T₁ - λ. But σ(T₁ - λ) = {0}, it follows from Lemma 2.7 that T₁ - λ is nilpotent. Therefore T₁ - λ has finite ascent and descent. On the other hand, since T₂ - 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

$$T-\lambda =\left(\begin{array}{cc}{T}_1& 0\\ 0& T_2\end{array}\right),\text{where}{T}_1\text{is}\text{invertible}\text{and}{T}_2\text{is}\text{nilpotent}.$$

Since T is algebraically quasi-paranormal, it has SVEP. So T₁ and T₂ have both finite ascent. But T₁ is Weyl, hence T₁ 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

$$T-\lambda =\left(\begin{array}{cc}\hfill T_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill T_2\hfill \end{array}\right),\text{where}{T}_1\text{is}\text{invertible}\text{and}{T}_2\text{is}\text{nilpotent}.$$

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

Next, we claim that σ _BW (f(T)) = f(σ _BW (T)) for each f ∈ H(σ(T)). Since $T\in g\mathcal{W},T\in g\mathcal{B}$. 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

$${\sigma }_{BW}\left(f\left(T\right)\right)={\sigma }_D\left(f\left(T\right)\right)=f\left({\sigma }_D\left(T\right)\right)=f\left({\sigma }_{BW}\left(T\right)\right).$$

Since 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

$$\sigma \left(f\left(T\right)\right)\backslash E\left(f\left(T\right)\right)=f\left(\sigma \left(T\right)\backslash E\left(T\right)\right).$$

Since $T\in g\mathcal{W}$, we have

$$\sigma \left(f\left(T\right)\right)\backslash E\left(f\left(T\right)\right)=f\left(\sigma \left(T\right)\backslash E\left(T\right)\right)=f\left({\sigma }_{BW}\left(T\right)\right)={\sigma }_{BW}\left(f\left(T\right)\right),$$

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

Now suppose that 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 $\sigma \left(T^{*}\right)=\overline{\sigma \left(T\right)},\bar{\lambda }$ 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

$$T-\lambda =\left(\begin{array}{cc}\hfill T_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill T_2\hfill \end{array}\right),\text{where}{T}_1\text{is}\text{invertible}\text{and}{T}_2\text{is}\text{nilpotent}.$$

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

$${\sigma }_{BW}\left(f\left(T\right)\right)=f\left({\sigma }_{BW}\left(T\right)\right)\text{for}\text{every}f\in H\left(\sigma \left(T\right)\right).$$

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 $\lambda \in ℂ\backslash {\sigma }_{BW}\left(T\right)$ 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 $\lambda \in ℂ\backslash {\sigma }_{BW}\left(T\right)$. 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

$$T\in g\mathcal{W}\text{if}\text{and}\text{only}\text{if}T\in {\mathcal{P}}_1\left(\mathcal{H}\right).$$

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

$$N\left(T\right)\subseteq N{\left(T+N\right)}^p\text{and}N\left(T+N\right)\subseteq N\left(T^p\right).$$

(3.3.1)

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) = {μ₁,μ₂,...,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

$$\sigma \left(T+F\right)=\sigma \left(\left(T+F\right)|Y_i\right)\cup \sigma \left(\left(T+F\right)|Z_i\right).$$

Let 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 )^mg(μ) with g(μ _i ) = 0. Then 0 = h(F _i ) = (F - μ _i )^mg(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

$$T_i+F_i=\left(T_i+{\mu }_i\right)+\left(F_i-{\mu }_i\right)=T_i+N_i+{\mu }_i.$$

(3.4.1)

Since 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

$$T+F=\underset{i=1}{\overset{n}{\oplus }}\left(T_i+F_i\right)\text{has}\text{SVEP}\text{.}$$

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 λ₀ ∈ E(T + F). Fix $i\in \mathbb{N}$ such that 1 ≤ i ≤ n. Since the equality T _i + N _i - λ₀ + μ _i = T _i + F _i - λ₀ holds, we consider two cases:

Case I: Suppose that T _i - λ₀ + μ _i is invertible. Since N _i is quasi-nilpotent commuting with T _i - λ₀ + μ _i , it is clear that T _i + F _i - λ₀ is also invertible. Hence H₀(T _i + F _i - λ₀) = N(T _i + F _i - λ₀) = {0}.

Case II: Suppose that T _i - λ₀ + μ _i is not invertible. Then λ₀ - μ _i ∈ σ(T _i ). We claim that λ₀ ∈ E(T _i + F _i ). Note that λ₀ ∈ σ(T _i + μ _i ) = σ(T _i + F _i ). Since σ(T _i + F _i ) ∈ σ(T + F) and λ₀ ∈ iso σ(T + F), λ₀ ∈ iso σ(T _i + N _i + μ _i ). Therefore λ₀ -μ _i ∈ iso σ(T _i + N _i ) = iso σ(T _i ). Since T _i - λ₀ + μ _i is algebraically quasi-paranormal, λ₀ - 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, λ₀ - 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

$$E\left(T_i\right)=\sigma \left(T_i\right)\backslash {\sigma }_{BW}\left(T_i\right)=\sigma \left(T_i+N_i\right)\backslash {\sigma }_{BW}\left(T_i+N_i\right).$$

Hence T _i + F _i - λ₀ is B-Weyl. Assume to the contrary that T _i + F _i - λ₀ is injective. Then β(T _i + F _i - λ₀) = α(T _i + F _i - λ₀) = 0. Therefore T _i + F _i - λ₀ is invertible, and so λ₀ ∉ σ(T _i +F _i ). This is a contradiction. Hence λ₀ ∈ 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 - λ₀) = N(T _i + F _i - λ₀) ^mi .

From Cases I and II we have

$$\begin{array}{ll}\hfill H_0\left(T+F-{\lambda }_0\right)& =\underset{i=1}{\overset{n}{\oplus }}{H}_0\left(T_i+F_i-{\lambda }_0\right)\\ =\underset{i=1}{\overset{n}{\oplus }}N{\left(T_i+F_i-{\lambda }_0\right)}^{m_i}\\ =N{\left(T+F-{\lambda }_0\right)}^m,\end{array}$$

where m : = max{m₁,m₂,...,m _n }. Since the last equality holds for every λ₀ ∈ 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 Fⁿ 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 Fⁿ is a finite-dimensional operator for some $n\in \mathbb{N}$. Then $T+F\in g\mathcal{W}$.