On operators satisfying an inequality

An operator T is said to be k-quasi-∗-class A if , where k is a natural number. Let denote either the generalized derivation or the elementary operator , where and are the left and right multiplication operators defined on by and respectively. This article concerns some spectral properties of k-quasi-∗-class A operators in a Hilbert space, as the property of being hereditarily polaroid. We also establish Weyl-type theorems for T and , where T is a k-quasi-∗-class A operator and A, are also k-quasi-∗-class A operators.

MSC: 47B47, 47A30, 47B20, 47B10.

Let be the algebra of all bounded linear operators acting on an infinite dimensional separable complex Hilbert space H. As an easy extension of normal operators, hyponormal operators have been studied by many mathematicians. Although there are many unsolved interesting problems for hyponormal operators (e.g., the invariant subspace problem), one of recent trends in operator theory is studying natural extensions of hyponormal operators. So, we introduce some of these non-hyponormal operators. Recall [1,2] that is called hyponormal if , and T is called paranormal (resp., ∗-paranormal) if (resp. ) for all unit vector . Following [2] and [3], we say that belongs to the class A if . Recently Jeon and Kim [3] have considered the following new class of operators: we say that an operator belongs to the ∗-class A if .

For brevity, we shall denote the classes of hyponormal operators, paranormal operators, ∗-paranormal operators, class A operators, and ∗-class A operators by ℋ, , , and respectively. From [1] and [2], it is well known that

Recently in [4], the authors have extended ∗-class A operators to quasi-∗-class A operators. An operator is said to be quasi-∗-class A if . If we denote this class of operators by , then

As a further generalization of both ∗-class A operators and quasi-∗-class A operators, the author in [5] introduced k-quasi-∗-class A operators. An operator T is called k-quasi-∗-class A if

where k is a natural number. Let be the class of k-quasi-∗-class A operators. Thus,

The spectral properties of quasi-class A and quasi-∗-class A operators have been investigated by many authors in the recent years (a useful survey on the spectral properties of these operators may also be found in [6]); see also [4]. In this paper we extend to k-quasi-∗-class A operators some of these results, for instance the property of being hereditarily polaroid already observed for ∗-paranormal operators and ∗-class A operators defined on Hilbert spaces [5].

The fine structure of the spectrum of paranormal operators for class A operators or ∗-paranormal operators has been studied by several authors, in particular, for these classes of operators, it has been proved that they satisfy Weyl’s theorem; see for instance [7,8] for paranormal operators, [9] for algebraically class A operators, in [5] for quasi-∗-class A operators. In this paper we extend these results by proving that some other variants of Weyl’s theorem hold for k-quasi-∗-class A operators; for instance, the so-called property (w) introduced by Rakočević in [10] and studied in [11] and [12]. All Weyl-type theorems are established for T and for ; T is a k-quasi-∗-class A operator and A, are also k-quasi-∗-class A operators.

We begin by explaining the relevant terminology. Let X be a complex Banach space. For a bounded linear operator , let denote the null space and ranT denote the range of T. Let be the ascent of an operator T. (I.e., the smallest non-negative integer p such that . If such integer does not exist, we put .) Analogously, let be the descent of an operator T; i.e., the smallest non-negative integer q such that , and if such integer does not exist, we put . It is well known that if and are both finite, then [[13], Proposition 38.3]. Moreover, precisely when λ is a pole of the resolvent of T; see Proposition 50.2 of Heuser [13]. A bounded operator is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. is said to be hereditarily polaroid if the restriction of T to any closed invariant subspace is polaroid. Let denote the classical approximate point spectrum. T is said to be a-polaroid if every is a pole of the resolvent of T. Obviously,

In [14] it has been observed that if the dual has SVEP (respectively, T has SVEP), then two conditions for T of being polaroid or a-polaroid (respectively, for ) are equivalent. The following property has a relevant role in local spectral theory and Fredholm operator theory; see the recent monographs by Laursen and Neumann [15] and [16]. A bounded operator is said to have the single valued extension property (abbreviated SVEP) if, for every open subset G of ℂ and any analytic function such that on G, we have on G.

We also have

and dually, if denotes the dual of T,

see [[16], Theorem 3.8]. In the case of Hilbert space operators, the last implication is still true if we replace with the Hilbert adjoint . A bounded operator is said to have Bishop’s property (β) if for every open subset G of ℂ and every sequence of H-valued analytic functions such that converges uniformly to 0 in norm on compact subsets of G, converges uniformly to 0 in norm on compact subsets of G. It is known that the property (β) for T entails that T has SVEP; see [15] for details.

We begin by the following lemma which is the essence of this paper and it is a structure theorem of a k-quasi-∗-class A operator T.

Lemma 2.1[5]

Letbe ak-quasi-∗-classAoperator, the range ofbe not dense and

Thenis a ∗-classAoperator, and.

As a consequence, we obtain the following corollary.

Corollary 2.1Letbe ak-quasi-∗-classAoperator. Ifis invertible, thenTis similar to a direct sum of a ∗-classAoperator and a nilpotent operator.

Proof Since by assumption we have , then there exists an operator S such that [17]. Hence,

 □

Corollary 2.2LetTbe ak-quasi-∗-classAoperator. IfTis quasinilpotent, then it must be a nilpotent operator.

Proof Invoking Lemma 2.1, we find . Since is ∗-class A, we conclude that [18]. Since , a computation shows that

 □

Lemma 2.2[5]

LetMbe a closedT-invariant subspace ofH. Then the restrictionof ak-quasi-∗-classAoperatorTtoMis ak-quasi-∗-classAoperator.

Theorem 2.1[5]

Letbek-quasi-∗-classA. ThenTsatisfies Bishop’s property (β), the single valued extension property and the Dunford property (C).

Lemma 2.3Letbe an algebraicallyk-quasi-∗-classAoperator, and, thenis nilpotent.

Proof Assume is k-quasi-∗-class A for some nonconstant polynomial . Since , the operator is nilpotent by Corollary 2.2. Let

where for . Then

and hence . □

In the following theorem, we will prove that an algebraically k-quasi-∗-class A operator is polaroid.

Theorem 2.2LetTbe an algebraicallyk-quasi-∗-classAoperator. ThenTis polaroid.

Proof If T is an algebraically k-quasi-∗-class A operator, then is a k-quasi-∗-class A operator for some nonconstant polynomial p. Let , and let be the Riesz idempotent associated to μ defined by

where D is a closed disk centered at μ which contains no other points of the spectrum of T. Then T can be represented as follows:

where and . Since is algebraically k-quasi-∗-class A operator by Lemma 2.3 and , it follows from Lemma 2.3 that is nilpotent. Therefore, has finite ascent and descent. On the other hand, since is invertible, it has finite ascent and descent. Therefore, has finite ascent and descent. Therefore, μ is a pole of the resolvent of T. Now if , then . Thus, , where denotes the set of poles of the resolvent of T. Hence, T is polaroid. □

Recall that an operator T is said to be hereditarily polaroid if every part of it is polaroid. Hence, it follows from Lemma 2.2 that a k-quasi-∗-class A operator is hereditarily polaroid

Corollary 2.3Ak-quasi-∗-classAoperator is isoloid.

Let X be a complex Banach space. For every , define

and

Obviously, for every . Define

and

Let , i.e., is the set of all poles of the resolvent of T.

Definition 3.1 A bounded operator is said to satisfy Weyl’s theorem, in symbol (W), if . T is said to satisfy a-Weyl’s theorem, in symbol (aW), if . T is said to satisfy the property (w), if .

Either a-Weyl’s theorem or the property (w) entails Weyl’s theorem. The property (w) and a-Weyl’s theorem are independent; see [11].

The concept of semi-Fredholm operators has been generalized by Berkani [19,20] in the following way: for every and a nonnegative integer n, let us denote by the restriction of T to viewed as a map from the space into itself (we set ). is said to be semi-B-Fredholm (resp. B-Fredholm, upper semi-B-Fredholm, lower semi-B-Fredholm,) if for some integer , the range is closed and is a semi-Fredholm operator (resp. Fredholm, upper semi-Fredholm, lower semi-Fredholm). In this case is a semi-Fredholm operator for all [20]. This enables one to define the index of a semi-B-Fredholm as . A bounded operator is said to be B-Weyl (respectively, upper semi-B-Weyl, lower semi-B-Weyl) if for some integer , is closed and is Weyl (respectively, upper semi-Weyl, lower semi-Weyl). In an obvious way, all the classes of operators generate spectra, for instance, the B-Weyl spectrum and the upperB-Weyl spectrum. Analogously, a bounded operator is said to be B-Browder (respectively, upper semi-B-Browder, lower semi-B-Browder) if for some integer , is closed and is Weyl (respectively, upper semi-Browder, lower semi-Browder). The B-Browder spectrum is denoted by , the upper semi-B-Browder spectrum by .

The generalized versions of Weyl-type theorems are defined as follows.

Definition 3.2 A bounded operator is said to satisfy generalized Weyl’s theorem, in symbol, (gW), if . is said to satisfy generalizeda-Weyl’s theorem, in symbol, (gaW), if . is said to satisfy the generalized property (w), in symbol, , if .

In the following diagrams, we resume the relationships between all Weyl-type theorems:

see [[21], Theorem 2.3], [11] and [22]. The generalized property (w) and generalized a-Weyl’s theorem are also independent; see [21]. Furthermore,

see [21] and [22]. The converse of all these implications in general does not hold. Furthermore, by [[23], Theorem 3.1],

(W) holds for T ⇔ Browder’s theorem holds for T and .

Let denote either the generalized derivation or the elementary operator , where and are the left and right multiplication operators defined on by and respectively. We will show that if A, are k-quasi-∗-class A, then is polaroid and satisfies all Weyl-type theorems. For this we need the following lemmas.

Lemma 3.1[24]

Let. IfA, Bare polaroid operators, thenis polaroid.

Lemma 3.2IfA, arek-quasi-∗-classAoperators, thenis polaroid.

Proof It is known in a Hilbert space [14] that B is polaroid if and only if is polaroid. Hence, it suffices to apply the previous lemma. □

Recall that an operator is said to have the property (δ) if for every open covering of ℂ, we have .

Lemma 3.3Let. IfA, Bhave the property (β), thenhas SVEP.

Proof It is known [[15], Theorem 2.5.5] that B satisfies the property (β) if and only if satisfies the property (δ). Since A, B have the property (β) by Theorem 2.1, satisfies the property (δ). Hence, it results from [[15], Corollary 3.6.16] that both and satisfy the Dunford property (C). Since and commute, hence and have SVEP by [[15], Theorem 3.6.3 and Note 3.6.19]. Therefore, satisfies SVEP. □

Corollary 3.1Let. IfA, arek-quasi-∗-classAoperators, thenhas SVEP.

If a Banach space operator T has SVEP (everywhere), the single-valued extension property, then T and satisfy Browder’s (equivalently, generalized Browder’s) theorem and a-Browder’s (equivalently, generalized a-Browder’s) theorem. A sufficient condition for an operator T satisfying Browder’s (generalized Browder’s) theorem to satisfy Weyl’s (resp. generalized Weyl’s) theorem is that T is polaroid. Now since T and are polaroid operators when T is algebraically k-quasi-∗-class A , then Weyl’s theorem and generalized Weyl’s theorem hold for T and when T is algebraically k-quasi-∗-class A. Now, observe for polaroid operators T satisfying generalized Weyl’s theorem,

where is the set of poles of the resolvent of T. Hence, for a polaroid operator T, satisfies generalized Weyl’s theorem if and only if T satisfies generalized Weyl’s theorem if and only if T satisfies Weyl’s theorem if and only if satisfies Weyl’s theorem.

Theorem 3.1Let. IfT, A, are algebraicallyk-quasi-∗-classA, then the following statements are equivalent.

(i) generalized Weyl’s theorem holds for (resp. for).

(ii) generalized Weyl’s theorem holds forT (resp. for).

(iii) Weyl’s theorem holds forT (resp. for).

Recall that a sufficient condition for an operator T satisfying Browder’s (generalized Browder’s) theorem to satisfy Weyl’s (resp. generalized Weyl’s) theorem is that T is polaroid. Observe that if has SVEP, then . Hence, if T has SVEP and is polaroid, then satisfies generalized a-Weyl’s (so, also a-Weyl’s) theorem [14]. It follows from Theorem 2.1 that k-quasi-∗-class A operator has SVEP. Thus, we have the following theorem.

Theorem 3.2Let.

(i) IfTis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then generalizeda-Weyl’s theorem holds for (resp. for).

(ii) Ifis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then generalizeda-Weyl’s theorem holds forT (resp. for).

Recall [14] that if T is polaroid, then T satisfies generalized Weyl’s theorem (resp. generalized a-Weyl’s theorem) if and only if T satisfies Weyl’s theorem (resp. a-Weyl’s theorem). Hence if T is an algebraically k-quasi-∗-class A operator, we have the following result.

Theorem 3.3Let. IfTis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then

(i) Weyl’s theorem holds forT (resp. for) if and only if generalized Weyl’s theorem holds forT (resp. for).

(ii) a-Weyl’s theorem holds for (resp. for) if and only if generalizeda-Weyl’s theorem holds for (resp. for).

Theorem 3.4Let.

(i) Ifis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem and generalizeda-Weyl’s theorem hold forT (resp. ) and these are equivalent.

(ii) IfTis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem and generalizeda-Weyl’s theorem hold for (resp. ) and are equivalent.

Proof Since Weyl’s theorem holds for T (resp. for ). It suffices to show that Weyl’s theorem is equivalent to each one of the other Weyl-type theorems for T (resp. for ), generalized or not. Since (resp. ) has SVEP, Weyl’s theorem and a-Weyl’s theorem hold for T (resp. for ) and are equivalent by [[8], Theorem 2.16]. Theorem 3.3(i) implies that Weyl’s theorem and generalized Weyl’s theorem hold for T (resp. for ) and are equivalent. Now a-Weyl’s theorem and generalized a-Weyl’s theorem hold for T (resp. for ) and are equivalent by Theorem 3.3(ii). □

Let , where is the space of all functions that are analytic in an open neighborhoods of . If T is polaroid, then is polaroid too [14]. Thus, we have

Theorem 3.5Let.

(i) Ifis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then (resp. ) satisfies Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem and generalizeda-Weyl’s theorem.

(ii) IfTis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then (resp. ) satisfies Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem and generalizeda-Weyl’s theorem.

Proof

(i) If is algebraically k-quasi-∗-class A and A, are k-quasi-∗-class A, then (resp. ) is polaroid [14]. Since (resp. ) is polaroid, the result holds by [[14], Theorem 3.12]

(ii) If T is algebraically k-quasi-∗-class A and A, are k-quasi-∗-class A , then (resp. ) is polaroid. Since T (resp. ) is polaroid, the result holds by [[14], Theorem 3.12].

 □

According to [[14], Theorem 3.12] Theorem 3.4 may be extended as follows.

Theorem 3.6Let.

(i) Ifis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem and generalizeda-Weyl’s theorem hold for (resp. ) and these are equivalent.

(ii) IfTis algebraicallyk-quasi-∗-classAandA, arek-quasi-∗-classA, then Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem and generalizeda-Weyl’s theorem hold for (resp. ) and these are equivalent.

Remark 3.1 According to [14], the previous results on a Weyl-type theorem still true for the property (w).

The author declares that they have no competing interests.

The author would like to thank the referee for his good reading of the paper and his comments. This paper is supported by Taibah University Research Center Project (1433-808).

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