Weyl-type theorems and k-quasi-M-hyponormal operators
© Zuo and Zuo; licensee Springer. 2013
Received: 24 April 2013
Accepted: 29 August 2013
Published: 1 November 2013
In this paper, we show that if E is the Riesz idempotent for a non-zeroisolated point λ of the spectrum of ak-quasi-M-hyponormal operator T, then E isself-adjoint, and . Also, we obtain that Weyl-type theorems hold foralgebraically k-quasi-M-hyponormal operators.
MSC: 47B20, 47A10.
Let T be a bounded linear operator on a complex Hilbert space H,write it for , take a complex number λ in ℂ, and,henceforth, shorten to . One of recent trends in operator theory is studying naturalextensions of normal operators. We introduce some of these operators as follows.
T is said to be a hyponormal operator if ;
where k is a natural number.
It is clear that .
We give the following example to indicate that there exists an M-hyponormaloperator, which is not hyponormal.
Example 1.1 Consider the unilateral weighted shift operator as aninfinite-dimensional Hilbert space operator. Recall that given a bounded sequence ofpositive numbers α: (called weights), the unilateral weighted shift associated with α is the operator on defined by for all , where is the canonical orthogonal basis for . It is well known that is hyponormal if and only if α is monotonicallyincreasing. Also, is M-hyponormal if and only if α iseventually increasing. Hence, if we take the weights α such that , , , , then is an M-hyponormal operator, but it is nothyponormal.
Next, we give a 2-quasi-M-hyponormal operator, which is notM-hyponormal.
Example 1.2 Let defined on . Then by simple calculations, we see that T is a2-quasi-M-hyponormal operator, but is not M-hyponormal.
If , we shall write and for the null space and the range space of T. Also, let , , and for the spectrum and the isolated points of the spectrum ofT, respectively.
Let . The Riesz idempotent E of T with respect toλ is defined by , where D is a closed disk, centered at λ,which contains no other points of . It is well known that the Riesz idempotent satisfies , , , and . Stampfli  showed that if T satisfies the growth condition , then E is self-adjoint and . Recently, Chō and Tanahashi  obtained an improvement of Stampfli’s result top-hyponormal operators or log-hyponormal operators. Furthermore, Chōand Han extended it to M-hyponormal operators as follows.
Proposition 1.3 [, Theorem 4]
Let T be an M-hyponormal operator, and let λ be an isolated point of. If E is the Riesz idempotent for λ, then E is self-adjoint, and.
2 Isolated point of spectrum of k-quasi-M-hyponormaloperators
Lemma 2.1 Let T be a k-quasi-M-hyponormal operator. If, and assume that, then.
Proof If and , then T is invertible, so T is anM-hyponormal operator, and hence, by . □
Lemma 2.2 Let T be a k-quasi-M-hyponormal operator and. Thenimplies that.
Proof Suppose that . Since T is a k-quasi-M-hyponormaloperator, for all vectors and . In particular, . Since , . , therefore . □
3 Weyl-type theorems of algebraically k-quasi-M-hyponormaloperators
We say that T is an algebraically k-quasi-M-hyponormaloperator if there exists a nonconstant complex polynomial p such that is a k-quasi-M-hyponormal operator. From thedefinition above, T is an algebraicallyk-quasi-M-hyponormal operator, then so is for each .
An operator T is called Fredholm if is closed, and both and are finite-dimensional. The index of a Fredholm operatorT is given by . An operator T is called Weyl if it is Fredholm of indexzero. The Weyl spectrum of T is defined by . Following , we say that Weyl’s theorem holds for T if , where .
More generally, Berkani investigated the B-Fredholm theory (see [10–12]). We define if there exists a positive integer n such that is closed, is upper semi-Fredholm (i.e., is closed, ) and . We define . Let denote the set of all isolated points λ of with . We say that generalized a-Weyl’s theorem holds forT if .
We know that Weyl’s theorem holds for hermitian operators , which have been extended to hyponormal operators , algebraically hyponormal operators by , algebraically M-hyponormal operators  and algebraically quasi-M-hyponormal operators , respectively. In this section, we obtain that generalizeda-Weyl’s theorems hold for algebraicallyk-quasi-M-hyponormal operators.
Thenis M-hyponormal, and.
Theorem 3.2 Let T be a quasinilpotent algebraically k-quasi-M-hyponormal operator. Then T is nilpotent.
where is an M-hyponormal operator. Since and , the operator is quasinilpotent. But , thus . So , and hence, is quasinilpotent. Since is a k-quasi-M-hyponormal operator, by theprevious argument is nilpotent. On the other hand, since , for some natural number m. . is nilpotent, therefore, T isnilpotent. □
Recall that an operator T is said to be isoloid if every isolated point of is an eigenvalue of T and polaroid if every isolatedpoint of is a pole of the resolvent of T. In general, ifT is polaroid, then it is isoloid. However, the converse is not true.In , it is showed that every algebraically M-hyponormal operator isisoloid, we can prove more.
Theorem 3.3 Let T be an algebraically k-quasi-M-hyponormal operator. Then T is polaroid.
where and . Since is an algebraically k-quasi-M-hyponormaloperator, so is . But , it follows from Theorem 3.2 that is nilpotent, thus has finite ascent and descent. On the other hand, since is invertible, clearly, it has finite ascent and descent. has finite ascent and descent, and hence, λ is apole of the resolvent of T, therefore, T ispolaroid. □
Corollary 3.4 Let T be an algebraically k-quasi-M-hyponormal operator. Then T is isoloid.
is a zero function on U.
Theorem 3.5 Let T be an algebraically k-quasi-M-hyponormal operator. Then T has SVEP.
Proof Suppose that T is an algebraicallyk-quasi-M-hyponormal operator. Then is a k-quasi-M-hyponormal operator for somenonconstant complex polynomial p, and hence, has SVEP by [, Theorem 2.1]. Therefore, T has SVEP by [, Theorem 3.3.9]. □
In the following theorem, denotes the space of functions analytic in an open neighborhood of .
Theorem 3.6 Let T orbe an algebraically k-quasi-M-hyponormal operator. ThenWeyl’s theorem holds forfor every.
Proof Firstly, suppose that T is an algebraicallyk-quasi-M-hyponormal operator. We first show that Weyl’stheorem holds for T. Using the fact [, Theorem 2.2] that if T is polaroid, then Weyl’stheorem holds for T if and only if T has SVEP at points of . We have that T is polaroid by Theorem 3.3, andT has SVEP by Theorem 3.5. Hence, T satisfiesWeyl’s theorem.
Next, suppose that is an algebraically k-quasi-M-hyponormaloperator. Now we show that Weyl’s theorem holds for T. We use thefact [, Theorem 3.1] that if T or has SVEP, then Weyl’s theorem holds for T if andonly if . Since has SVEP, it is sufficient to show that . is clear, so we only need to prove . Let . Then λ is an isolated point of . Hence, λ is a pole of the resolvent of T,since T is polaroid by Theorem 3.3, that is, . By assumption, we have , so . Hence, we conclude that . Therefore, Weyl’s theorem holds for T.
Finally, we can derive the result by Theorem 3.5 and [, Theorem 2.4]. □
Following [, Theorem 3.12], we obtain the following result.
If is an algebraically k-quasi-M-hyponormal operator, then satisfies a generalized a-Weyl’s theorem.
If T is an algebraically k-quasi-M-hyponormal operator, then satisfies a generalized a-Weyl’s theorem.
We wish to thank the referee for careful reading and valuable comments for theorigin draft. This work is supported by the Basic Science and TechnologicalFrontier Project of Henan Province (No. 132300410261). This work is partiallysupported by the National Natural Science Foundation of China (No. 11271112,11201126).
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