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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 ( H ) and B 0 ( H ) denote, respectively, the algebra of bounded linear operators and the ideal of compact operators acting on . If TB ( H ) 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 TB ( H ) 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 TB ( H ) is either upper or lower semi-Fredholm, then T is called semi-Fredholm, and index of a semi-Fredholm operatorTB ( H ) is defined by

i ( T ) : = α ( T ) - β ( T ) .

If both α(T) and β(T) are finite, then T is called Fredholm. TB ( H ) is called Weyl if it is Fredholm of index zero. For TB ( H ) 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 BF ( H ) 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 TB ( H ) 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 xA is Drazin invertible of degree k if there exists an element aA such that

x k a x = x k , a x a = a , and x a = a x .

Let aA. Then the Drazin spectrum is defined by

σ D ( a ) : = { λ : a - λ is not Drazin invertible } .

For TB ( H ) , the smallest nonnegative integer p such that N (Tp ) = 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(Tq ) = R(Tq+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 T 2 , where T 1 is invertible and T 2 is nilpotent .

An operator TB ( H ) 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

σ B F ( T ) : = { λ : T - λ is not B - Fredholm } ,
σ B W ( T ) : = { λ : T - λ is not B - Weyl } .

Now, we consider the following sets:

B F + ( ) : = { T B ( ) : T is upper semi- B -Ferdholm}, B F + ( ) : = { T B ( ) : T B F + ( ) and i ( T ) 0 }, L D ( ) : = { T B ( ) : p ( T ) < and R ( T p ( T ) + 1 ) is closed} .

By definition,

σ B e a ( T ) : = { λ : T - λ B F + - ( H ) } ,

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

σ L D ( T ) : = { λ : T - λ L D ( H ) }

is the left Drazin spectrum. It is well known that

σ B e a ( T ) σ L D ( T ) = σ B e a ( T ) acc σ a ( T ) σ D ( T ) ,

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

p 0 a ( T ) : = { λ σ α ( T ) : T - λ L D ( H ) } , π 0 a ( T ) : = { λ iso σ a ( T ) : λ σ p ( T ) } .

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 H which satisfies the equation

( T - λ ) f ( λ ) = 0

is the constant function f ≡ 0 on U. T has SVEP if T has SVEP at every point λ.

Definition 1.1. Let TB ( H ) .

  1. (1)

    Generalized Weyl's theorem holds for T (in symbols, TgW) if

    σ ( T ) \ σ B W ( T ) = E ( T ) .
  2. (2)

    Generalized Browder's theorem holds for T (in symbols, TgB) if

    σ ( T ) \ σ B W ( T ) = π ( T ) .
  3. (3)

    Generalized a-Weyl's theorem holds for T (in symbols, T g a W ) if

    σ a ( T ) \ σ B e a ( T ) = π 0 a ( T ) .
  4. (4)

    Generalized a-Browder's theorem holds for T (in symbols, TgaB) if

    σ a ( T ) \ σ B e a ( T ) = p 0 a ( T ) .

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

g a Weyl's theorem g a Browder's theorem g Weyl ' s theorem g Browder ' s theorem

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 TB ( H ) is said to be class A if

T 2 T 2 .
  1. (2)

    T is called a quasi-class A operator if

    T * T 2 T T * T 2 T .
  2. (3)

    An operator TB ( H ) is said to be paranormal if

    T x 2 T 2 x x for all x 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 TB ( H ) is called quasi-paranormal if

T 2 x 2 T 3 x T x for all x H .

We say that TB ( H ) 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 TB ( H ) is quasi-paranormal if and only if

T * ( T 2 * T 2 - 2 λ T * T + λ 2 ) T 0 for all λ > 0 .

Example 2.4. T = I 0 I 0 B ( 2 2 ) . Then it is quasi-paranormal but not quasi-class

A.

Proof. Since T * = I I 0 0 , T 2 = ( T * ) 2 T 2 = I I 0 0 2 I 0 I 0 2 = 2 I 0 0 0

Therefore T * T 2 T = I I 0 0 2 I 0 0 0 I 0 I 0 = 2 I 0 0 0

On the other hand, since T 2 = T * T = I I 0 0 I 0 I 0 = 2 I 0 0 0 ,

T * T 2 T = I I 0 0 2 I 0 0 0 I 0 I 0 = 2 I 0 0 0 . Hence T is not quasi-class A.

However, since

T 2 * T 2 - 2 λ T * T + λ 2 = ( 2 - 4 λ + λ 2 ) I 0 0 λ 2 I ,

we have

T * ( T 2 * T 2 - 2 λ T * T + λ 2 ) T = 2 ( 1 - λ ) 2 I 0 0 0 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 = ( I 0 I I ) B ( 2 2 ) . Then it is algebraically quasi-paranormal but not quasi-paranormal.

Proof. Since T * = ( I I 0 I ) , we have

T 2 * T 2 2 λ T * T + λ 2 = ( ( λ 2 4 λ + 5 ) I ( 2 λ + 2 ) I ( 2 λ + 2 ) I ( λ 2 2 λ + 1 ) I ) .

Therefore

T * ( T 2 * T 2 2 λ T * T + λ 2 ) T = ( ( 2 λ 2 10 λ + 10 ) I ( λ 2 4 λ + 3 ) I ( λ 2 4 λ + 3 ) I ( λ 2 2 λ + 1 ) I ) .

Since (2λ2 - 10λ + 10)I is not a positive operator for λ = 2, T * ( T 2 * T 2 - 2 λ T * T + λ 2 ) T0 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]:

  1. (i)

    If TB ( H ) is algebraically quasi-paranormal, then so is T-λ for each λ.

  2. (ii)

    If TB ( H ) is algebraically quasi-paranormal and is a closed T-invariant subspace

of , then T|M is algebraically quasi-paranormal.

  1. (iii)

    If T is algebraically quasi-paranormal, then T has SVEP.

  2. (iv)

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

T is quasi-paranormal T = ( A B 0 0 ) on H = T H ¯ N ( T * ) ,

where A=T| T H ¯ is paranormal.

An operator TB ( H ) is called isoloid if iso σ(T) σ p (T) and an operator TB ( H ) is called polaroid if iso σ(T) π(T).

In general, the following implications hold:

T polaroid T isoloid .

However, each converse is not true. Consider the following example: let TB ( 2 ) be defined by

T x 1 , x 2 , x 3 , = ( 1 2 x 2 , 1 3 x 3 , ) .

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 T : = x H : lim n T n x 1 n = 0 .

If TB ( H ) , then the analytic core K(T) is the set of all xH such that there exists a constant c > 0 and a sequence of elements x n H such that x0 = x, Tx n = x n -1, and ║x n ║≤ cn║x║ for all n, see [6] for information on K(T).

Let P H denotes the class of all operators for which there exists p:=p λ for which

H 0 T - λ = N T - λ p for all λ ,

and P 1 ( H ) denotes the class of all operators for which there exists p:=p λ for which

H 0 T - λ = N T - λ p for all λ E T .

Evidently, P H P 1 H . Now we give a characterization of P 1 ( H ) .

Theorem 2.6. T P 1 H if and only if π(T) = E(T).

Proof. Suppose T P 1 H and let λ E(T). Then there exists p 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

H = H 0 T - λ K T - λ = N T - λ p K T - λ .

Therefore, we have

T - λ p H = T - λ p K T - λ = K T - λ ,

and hence H=N T - λ p T - λ p H , 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 P 1 H . □

From Theorem 2.6, we can give a simple example which belongs to P 1 ( H ) but not P H . Let U be the unilateral shift on 2 and let T = U*. Then T does not have SVEP at 0, and so H0(T) is not closed. Therefore TP H . However, since σ T = D ̄ , π T = E T = , where is an open unit disk in . Hence T P 1 H 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 TB ( H ) 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 = ( A B 0 0 ) on = T ( ) ¯ N ( T * ) ,

where A:=T| T H ¯ 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 ( T ) = ( C D 0 0 ) on = p ( T ) ( ) ¯ N ( p ( T ) * ) ,

where C:=p T | p T H ¯ 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) = czm (z - λ1)(z - λ2) ... (z - λ n ) for some natural number m. Therefore p(T) = cTm (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 TB ( H ) be algebraically quasi-paranormal. Then T P 1 H .

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:= 1 2 π i D μ - T - 1 dμ, 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 = ( T 1 0 0 T 2 ) , where σ ( T 1 ) = { λ } and σ ( T 2 ) = σ ( T ) \ { λ } .

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 P 1 H .

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 T gW for each f H(σ(T)).

Proof. Suppose T is algebraically quasi-paranormal. We first show that TgW. 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 λ = ( T 1 0 0 T 2 ) , where T 1 is invertible and T 2 is 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 P 1 H . 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 - λ = T 1 0 0 T 2 , where T 1 is invertible and T 2 is 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 TgW.

Next, we claim that σ BW (f(T)) = f(σ BW (T)) for each f H(σ(T)). Since TgW,TgB. 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 T gB by [5, Theorem 2.9], and so σ BW (f(T)) = σ D (f(T)). Therefore we have

σ B W f T = σ D f T = f σ D T = f σ B W T .

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

σ f T \ E f T = f σ T \ E T .

Since TgW, we have

σ f T \ E f T = f σ T \ E T = f σ B W T = σ B W f T ,

which implies that f T gW.

Now suppose that T* is algebraically quasi-paranormal. We first show that TgW. Let λ σ(T) \ σ BW (T). Observe that σ T * = σ T ¯ and σ B W T * = σ B W T ¯ . So λ ¯ σ T * \ σ B W T * , and so λ ¯ E T * because T * gW. Since T* is algebraically quasi-paranormal, it follows from Theorem 2.8 that λ ¯ π T * . 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 σ T * = σ T ¯ , λ ̄ is an isolated point of σ(T*). Since T* is isoloid, λ ¯ E T * . 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 - λ = T 1 0 0 T 2 , where T 1 is invertible and T 2 is 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 TgW. If T* is algebraically quasi-paranormal then T is isoloid. It follows from the first part of the proof that f T gW. 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

σ B W f T = f σ B W T for every f H σ T .

An operator XB H is called a quasiaffinity if it has trivial kernel and dense range. SB H is said to be a quasiaffine transform of TB ( H ) (notation: S T) if there is a quasiaffinity XB H 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 ( S ) gaB 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 ( S ) gaB 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 TB ( H ) . Then the following statements are equivalent:

  1. (1)

    TgW;

  2. (2)

    T has SVEP at every λ \ σ B W T and π(T) = E(T).

Proof. Observe that TgB if and only if σ BW (T) = σ D (T). So TgB if and only if T has SVEP at every λ \ σ B W T . Therefore we obtain the desired conclusion. □

From this lemma, we obtain the following corollary

Corollary 3.2. Let TB ( H ) . Suppose T has SVEP. Then

T g W if and only if T P 1 H .

Proof. Since T has SVEP, TgBby Lemma 3.1. So σ(T) \ σ BW (T) = π(T). Therefore TgW if and only if T P 1 H by Theorem 2.6. □

Lemma 3.3. Suppose TB ( H ) and N is nilpotent such that TN = NT. Then T P 1 H if and only if T+N P 1 ( H ) .

Proof. Suppose Np = 0 for some p. Observe that without any assumption on T we have

N T N T + N p and N T + N N T p .
(3.3.1)

Suppose now that T P 1 H , 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(Tp ) > 0. Therefore α(T) > 0, and hence 0 E(T). So the equality E(T) = E(T + N) is proved. Suppose T P 1 H . Then π(T) = E(T) by Theorem 2.6, and so π(T + N) = π(T) = E(T) = E(T + N). Therefore T+N P 1 ( H ) . Conversely, if T+N P 1 ( H ) 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+FgW.

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

σ T + F = σ T + F | Y i σ T + F | Z i .

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 = T i + μ i + F i - μ i = T i + N i + μ 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 = i = 1 n T i + F i has SVEP .

Now, we show that T+F P 1 ( H ) . Since T i + μ i is algebraically quasi-paranormal, T i + μ i P 1 Y i by Theorem 2.8. By Lemma 3.3 and (3.4.1), T i + F i P 1 Y i for every i = 1, 2,...,n. Now assume that λ0 E(T + F). Fix i 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 gW 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 gB. Therefore we have σ BW (T i + N i ) = σ D (T i + N i ). Hence

E T i = σ T i \ σ B W T i = σ T i + N i \ σ B W T i + N i .

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 P 1 Y i 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

H 0 T + F - λ 0 = i = 1 n H 0 T i + F i - λ 0 = i = 1 n N T i + F i - λ 0 m i = N T + F - λ 0 m ,

where m : = max{m1,m2,...,m n }. Since the last equality holds for every λ0 E(T + F), T+F P 1 ( H ) . Therefore T+FgWby Corollary 3.2. □

It is well known that if for an operator FB ( H ) there exists a natural number n for which Fn is finite-dimensional, then F is algebraic.

Corollary 3.5. Suppose TB ( H ) is algebraically quasi-paranormal and F is an operator commuting with T such that Fn is a finite-dimensional operator for some n. Then T+FgW.

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Acknowledgements

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 quasi-paranormal operators. J Inequal Appl 2012, 89 (2012). https://doi.org/10.1186/1029-242X-2012-89

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Keywords

  • algebraically quasi-paranormal operator
  • generalized Weyl's theorem
  • single valued extension property