Abstract
An operator is called quasi-class if for a positive integer , which is a common generalization of class . In this paper, firstly we consider some spectral properties of quasi-class operators; it is shown that if is a quasi-class operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of are mutually orthogonal, and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam's theorems hold for class operators. Particularly, we show that if is a class operator and either or is not connected, then has a nontrivial invariant subspace.
1. Introduction
Throughout this paper let be a separable complex Hilbert space with inner product . Let denote the -algebra of all bounded linear operators on . The spectrum of an operator is denoted by .
Here an operator is called -hyponormal for if ; when , is called hyponormal; when , is called semihyponormal. is called log-hyponormal if is invertible and . And an operator is called paranormal if for all . By the celebrated Löwner-Heinz theorem “ ensures for any ,” every -hyponormal operator is -hyponormal for . And every invertible -hyponormal operator is log-hyponormal since logt is an operator monotone function. We remark that as for positive invertible operator , so that -hyponormality of approaches log-hyponormality of as . In this sense, log-hyponormal can be considered as 0-hyponormal. -hyponormal, log-hyponormal, and paranormal operators were introduced by Aluthge [1], Tanahashi [2], and Furuta [3, 4], respectively.
In order to discuss the relations between paranormal and -hyponormal and log-hyponormal operators, Furuta, et al. [5] introduced a very interesting class of bounded linear Hilbert space operators: class defined by , where which is called the absolute value of , and they showed that class is a subclass of paranormal and contains -hyponormal and log-hyponormal operators. Class operators have been studied by many researchers, for example, [6–12].
Aluthge [1] introduced , which is called Aluthge transformation of . The operator plays an important role in the study of spectral properties of the -hyponormal or log-hyponormal operator . Aluthge-Wang [13] introduced -hyponormal operators defined by where the polar decomposition of is and is the Aluthge transformation of . As a generalization of -hyponormal, Ito [14] introduction class is defined by for and . Ito and Yamazaki [15] showed that -hyponormal equals ; class equals . Inclusion relations among these classes are known as follows:
Jeon and Kim [16] introduced quasi-class (i.e., ) operators as an extension of the notion of class operators.
Recently Tanahashi et al. [9] considered an extension of quasi-class operators, similar with respect to the extension of the notion of -quasihyponormality to -quasihyponormality.
Definition 1.1. is called a quasi-class operator for a positive integer if
Remark 1.2. In [17], this class of operators is called -quasi-class . It is clear that
In [17], we show that the inclusion relation (1.6) is strict by an example.
In this paper, firstly we consider some spectral properties of quasi-class operators; it is shown that if is a quasi-class operator for a positive integer , then the nonzero points of its point spectrum and joint point spectrum are identical; furthermore, the eigenspaces corresponding to distinct eigenvalues of are mutually orthogonal; the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam’s theorems hold for class operators. Particularly, we show that if is a class operator and either or is not connected, then has a nontrivial invariant subspace.
2. Main Results
A complex number is said to be in the point spectrum of if there is a nonzero such that . If in addition, , then is said to be in the joint point spectrum of . Clearly, . In general, .
In [18], Xia showed that if is a semihyponormal operator, then ; Tanahashi extended this result to log-hyponormal operators in [2]. Aluthge [13] showed that if is -hyponormal, then nonzero points of and are identical; Uchiyama extended this result to class operators in [10]. In the following, we will point out that if is a quasi-class operator for a positive integer , then nonzero points of and are also identical and the eigenspaces corresponding to distinct eigenvalues of are mutually orthogonal.
Lemma 2.1 (see [9, 17]). Let be a quasi-class operator for a positive integer . If and for some , then .
Theorem 2.2. Let be a quasi-class operator for a positive integer . Then the following assertions hold:
(1) ,
(2) If , and , then .
Proof.
(1) Clearly by Lemma 2.1.
(2) Without loss of generality, we assume . Then we have by Lemma 2.1. Thus we have . Since , .
A complex number is said to be in the approximate point spectrum of if there is a sequence of unit vectors in such that . If in addition, , then is said to be in the joint approximate point spectrum of . Clearly, . In general, . In [18], Xia showed that if is a semihyponormal operator, then ; Tanahashi [2] extended this result to log-hyponormal operators. Aluthge and Wang [19] showed that if is -hyponormal, then nonzero points of and are identical; Chō and Yamazaki extended this result to class operators in [7]. In the following, we will show that if is a quasi-class operator for a positive integer , then nonzero points of and are also identical.
Theorem 2.3. Let be a quasi-class operator for a positive integer . Then .
To prove Theorem 2.3, we need the following auxiliary results.
Lemma 2.4 (see [20]). Let be a complex Hilbert space. Then there exists a Hilbert space such that and a map such that
(1) is a faithful -representation of the algebra on ;
(2) for any in ;
(3) for any .
Lemma 2.5 (see [18]). Let be Berberian’s faithful -representation. Then .
Proof of Theorem 2.3. Let be Berberian’s faithful -representation of Lemma 2.4. In the following, we shall show that is also a quasi-class operator for a positive integer .
In fact, since is a quasi-class operator, we have
Hence, we have
The proof is complete.
Theorem 2.6. Let be a quasi-class operator for a positive integer . Then (i.e., ).
Proof. It suffices to prove .
Xia [18] showed that holds for any . Hence we have
by Theorem 2.3. The proof is complete.
Putnam [21] proved three theorems concerning spectral properties of hyponormal operators. These theorems were generalized to -hyponormal operators by Chō et al. in [22, 23], to -hyponormal operators by Aluthge and Wang in [24], and to operators by Yang and Yuan in [25]. In the following, we extend these theorems to quasi-class operators.
We show the first generalization concerning points in the approximate point spectrum of a quasi-class operator for a positive integer as follows.
Theorem 2.7. Let be a quasi-class operator for a positive integer . If such that , then .
To prove Theorem 2.7, we need the following auxiliary results.
Lemma 2.8 (see [26]). Let be the polar decomposition of , , and a sequence of vectors. Then the following assertions are equivalent:
(1) and ,
(2) and ,
(3) and .
Proof of Theorem 2.7. If and , a sequence of unit vectors exists such that and by Theorem 2.3. Hence Theorem 2.7 holds by Lemma 2.8.
Corollary 2.9. Let be a class operator. If such that , then .
Let be a -hyponormal operator. Does it follow that if , then ? The answer is affirmative if by Corollary 2.9. In general, the answer is negative even if is hyponormal and the polar factor is unitary; see details in [21]. However, the converse is true for many classes of operators; see the following results.
Theorem 2.10 (see [18, 21, 23]). Let be -hyponormal for , then , where is defined by .
Indeed, the above Theorem 2.10 that was shown for the case is hyponormal by Putnam in [21], for the case is semihyponormal by Xia in [18], and the general case by Chō et al. in [23].
Theorem 2.11 (see [24]). Let be -hyponormal and is connected, then , where is defined by .
Here we show the second generalization concerning the relation between the spectrum of and to class operators with connected spectrum.
Theorem 2.12. Let be a class operator and is connected, then , where is defined by .
The numerical range of an operator is defined by Let denote the closure of . It is well known that for any , is a convex set and . Moreover, if is normal, then , the convex hull of .
We need the following auxiliary results.
Lemma 2.13 (see [7]). Let be the polar decomposition of a class operator and . Then is semihyponormal and
Lemma 2.14. Let be the polar decomposition of a class operator and . Then .
Proof. Let be the polar decomposition of . The nonzero points of and are identical. Since is a class operator, is semihyponormal by Lemma 2.13, that is, . It follows that if . Therefore . Hence
Lemma 2.15. Let be the polar decomposition of a class operator and . Then .
Proof. Since is a class operator, we have by the proof of Theorem 2.1 in [7]. So we have for any unit vector . By Lemma 2.14, . The convexity of and the above inequalities imply Hence
Proof of Theorem 2.12. Since is a class operator, is semihyponormal by Lemma 2.13. It follows from Theorem 2.10 that Since the nonzero points of and are identical, and implies that is not invertible, and hence , the above containment may be modified to become By Lemma 2.13, we have So Since is connected, is a closed convex subset of . Hence by Lemma 2.15, we have Since so we have that is, The proof is complete.
Putnam [21] proved that if is hyponormal and is not an interval, then has a nontrivial invariant subspace. This result has been generalized by many authors. Chō et al. generalized Putnam's result to -hyponormal operators in [22]. In [24], Aluthge and Wang proved that if is -hyponormal and either or is not an interval, then has a nontrivial invariant subspace.
Here we shall generalize the above result to class operators and give an application of Theorem 2.12.
A complex number is said to be in the compression spectrum of if is not dense in . It is well known that for any . Moreover, if and , then is a nontrivial invariant subspace of .
Theorem 2.16. Let be a quasi-class operator for a positive integer . If there is a , , with , then has a nontrivial invariant subspace.
Proof. We have that by Theorem 2.7. So we have . By the assumption, we have that . Hence has a nontrivial subspace.
Corollary 2.17. Let be a class operator. If there is a , , with , then has a nontrivial invariant subspace.
Theorem 2.18. Let be a class operator for a positive integer . If either or is not connected, then has a nontrivial invariant subspace.
Proof. We only give the proof for the case that is not connected, for the case is not connected can be proved similarly.
If is not connected, then the Theorem is clear, so we assume that is connected. By the assumption, we have that is not an interval, so there exist , , such that
Let . Since , there exists a for which . Similarly, if , then there exists a for which by Theorem 2.12.
On the other hand, if , then is not invertible and hence . Hence there exists a such that . So both the outer and inner boundaries of the annulus contain a point of . Since is connected, we have that .
Hence there exists a , thus . So we have that and by (2.19). Therefore Theorem 2.18 holds by Corollary 2.17.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (11071188) and the Natural Science Foundation of the Department of Education, Henan Province (2011A11009).