Open Access

Generalized weyl's theorem for algebraically quasi-paranormal operators

Journal of Inequalities and Applications20122012:89

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

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 operatorgeneralized Weyl's theoremsingle valued extension property

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 T B ( 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 T B ( 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 T B ( H ) is either upper or lower semi-Fredholm, then T is called semi-Fredholm, and index of a semi-Fredholm operator T B ( H ) is defined by
i ( T ) : = α ( T ) - β ( T ) .
If both α(T) and β(T) are finite, then T is called Fredholm. T B ( H ) is called Weyl if it is Fredholm of index zero. For T B ( H ) 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 T0 = 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 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 T B ( 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 x A is Drazin invertible of degree k if there exists an element a A such that
x k a x = x k , a x a = a , and x a = a x .
Let a A . Then the Drazin spectrum is defined by
σ D ( a ) : = { λ : a - λ is not Drazin invertible } .
For T B ( H ) , 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 T 2 , where T 1 is invertible and T 2 is nilpotent .
An operator T B ( 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 T B ( H ) .
  1. (1)
    Generalized Weyl's theorem holds for T (in symbols, T g W ) if
    σ ( T ) \ σ B W ( T ) = E ( T ) .
     
  2. (2)
    Generalized Browder's theorem holds for T (in symbols, T g B ) 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, T g a B ) 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 T B ( 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 T B ( 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 T B ( H ) is called quasi-paranormal if
T 2 x 2 T 3 x T x for all x H .

We say that T B ( 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 T B ( 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 ) T 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]:
  1. (i)

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

     
  2. (ii)

    If T B ( 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 T B ( H ) is called isoloid if iso σ(T) σ p (T) and an operator T B ( 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 T B ( 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 T B ( H ) , then the analytic core K(T) is the set of all x H 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 ║≤ c n ║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 T P 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 T B ( 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) = 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 B ( 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 d μ , 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 g W for each f H(σ(T)).

Proof. Suppose T is algebraically quasi-paranormal. We first show that T g 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 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 T g W .

Next, we claim that σ BW (f(T)) = f(σ BW (T)) for each f H(σ(T)). Since T g W , T g 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 T g B 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 T g W , 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 g W .

Now suppose that T* is algebraically quasi-paranormal. We first show that T g W . 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 * g W . 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 T g W . If T* is algebraically quasi-paranormal then T is isoloid. It follows from the first part of the proof that f T g 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
σ B W f T = f σ B W T for every f H σ T .

An operator X B H is called a quasiaffinity if it has trivial kernel and dense range. S B H is said to be a quasiaffine transform of T B ( H ) (notation: S T) if there is a quasiaffinity X B 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 ) g a 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 ( S ) g a 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 B ( H ) . Then the following statements are equivalent:
  1. (1)

    T g W ;

     
  2. (2)

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

     

Proof. Observe that T g B if and only if σ BW (T) = σ D (T). So T g B 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 T B ( H ) . Suppose T has SVEP. Then
T g W if and only if T P 1 H .

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

Lemma 3.3. Suppose T B ( 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 N p = 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(T p ) > 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 + F g 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
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 ) 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 + 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 g 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 g B . 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 + F g W by Corollary 3.2. □

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

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

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department Of Mathematics, College Of Sciences, Kyung Hee University

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© An and Han; licensee Springer. 2012

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