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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 461567, 7 pages
http://dx.doi.org/10.1155/2014/461567
Research Article

On Subscalarity of Some 2 × 2 M-Hyponormal Operator Matrices

1College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan, China
2School of Mathematical Science, Inner Mongolia University, Hohhot 010021, Inner Mongolia, China

Received 25 December 2013; Accepted 16 January 2014; Published 25 February 2014

Academic Editor: Yisheng Song

Copyright © 2014 Fei Zuo and Junli Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide some conditions for operator matrices whose diagonal entries are -hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.

1. Introduction and Preliminaries

Let be a complex separable Hilbert space and let denote the algebra of all bounded linear operators on . If , we write , , , and for the null space, the range space, the spectrum, and the approximate point spectrum of , respectively. An operator is called Fredholm if is closed, , and . The index of a Fredholm operator is given by . An operator is called Weyl if it is Fredholm of index zero. The Weyl spectrum of [1] is defined by .

We consider the sets and define where iso denotes the isolated points of .

Following [2], we say that Weyl’s theorem holds for if and that -Weyl’s theorem holds for if .

Let . As an easy extension of normal operators, hyponormal operators have been studied by many mathematicians. Though 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 nonhyponormal operators. An operator is said to be -hyponormal if there exists a real positive number such that Evidently,

There is a vast literature concerning -hyponormal operators (see [35], etc.). We also note that an operator need not be hyponormal even though and are both -hyponormal. To see this, consider the operator where is the unilateral shift on and is given by . Then a direct calculation shows that

for all and for all , which says that and are both -hyponormal. But since while is not hyponormal.

Let be the coordinate in the complex plane and let denote the planar Lebesgue measure. Fix a complex (separable) Hilbert space and a bounded (connected) open subset of . We will denote by the Hilbert space of measurable functions , such that

The Bergman space for is defined by , where denotes the Fréchet space of -valued analytic functions on with respect to uniform topology. Note that is a Hilbert space. Let us define now a special Sobolev type space. Let be again a bounded open subset of and let be a fixed nonnegative integer. The vector valued Sobolev space with respect to and of order will be the space of those functions whose derivatives in the sense of distributions still belong to . Endowed with the norm becomes a Hilbert space contained continuously in . A bounded linear operator on is called scalar of order if it possesses a spectral distribution of order , that is, if there is a continuous unital morphism of topological algebras such that , where stands for the identity function on , and stands for the space of compactly supported functions on , continuously differentiable of order . An operator is subscalar if it is similar to the restriction of a scalar operator to an invariant subspace. Let be a (connected) bounded open subset of and let be a nonnegative integer. The linear operator of multiplication by on is continuous and it has a spectral distribution of order , defined by the functional calculus Therefore, is a scalar operator of order .

An operator is said to have the single-valued extension property (or SVEP) if for every open subset of and any analytic function such that on , we have on .

An operator is said to have Bishop’s property if for every open subset of and every sequence of -valued analytic functions such that converges uniformly to 0 in norm on compact subsets of , converges uniformly to 0 in norm on compact subsets of . It is well known that

In 1984, Putinar showed in [6] that every hyponormal operator is subscalar, and then in 1987, Brown used this result to prove that a hyponormal operator with rich spectrum has a nontrivial invariant subspace (see [7]). There have been a lot of generalizations of such beautiful consequences (see [811]). In this paper, we provide some conditions for operator matrices whose diagonal entries are -hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.

2. Subscalarity

Lemma 1 (see [6, Proposition  2.1]). For a bounded open disk in the complex plane , there is a constant such that for an arbitrary operator and we have where denotes the orthogonal projection of onto the Bergman space .

Corollary 2. Let be as in Lemma 1. If is an -hyponormal operator, then there exists a constant such that for all and where denotes the orthogonal projection of onto the Bergman space .

Proof. This follows from Lemma 1 and the definition of -hyponormal operator.

Lemma 3. Let be an -hyponormal operator and let be a bounded disk in . If is a sequence in such that for , then for , where is a disk strictly contained in .

Proof. Since is an -hyponormal operator, it follows from Corollary 2 that there exists a constant such that for . From (17), we have for . Hence, for . Since has Bishop’s property [12], we have for , where denotes a disk with . From (18) and (20), we get that for .

Lemma 4. Let , where are mutually commuting, and both and are -hyponormal operators. For any positive integer and any bounded disk in containing , define the map by where and denotes the constant function sending any to . Then the following statements hold.(i)If for some nonnegative integer and , where , then is one-to-one and has closed range, where .(ii)If , , and is algebraic of order , then is one-to-one and has closed range.(iii)If is an -hyponormal operator and , then is one-to-one and has closed range.

Proof. Let and be sequences such that Then (25) implies that By the definition of the norm of Sobolev space and (26) we get
for .(i)Set  , where . We may assume that . Then .
We prove that for every , the following equations hold for , where .
To prove (28), we will use the induction on . Since , then (28) holds when . Suppose that (28) is true for some . From (27) and the inductions hypothesis, we have for . Since is an -hyponormal operator, by Lemma 3 we have
for , where . From (30) and the second equation of (27), for . Since is an -hyponormal operator, by Lemma 3 we derive for , where . Therefore, the proof of (28) is completed. Let in (28); we have for . From (27) and (33), it follows that for . Since is an -hyponormal operator, from Lemma 3 we obtain that for , where , and hence for . Since is an -hyponormal operator, Lemma 3 implies that for , where . By repeating the process from (33) to (37), it holds for all that for , where . In particular, let , for . Hence, from the first equation of (27), we have for . Applying Corollary 2, we have where denotes the orthogonal projection of onto and . By combining (26) with (39) and (41), we obtain that where .
Let be a curve in surrounding . Then for uniformly. Hence, by the Riesz functional calculus, But by Cauchy’s theorem. Hence, , and so is one-to-one and has closed range.(ii)Set . By the hypothesis and (27), we have for . Since is algebraic with order , there exists a nonconstant polynomial such that . Set for and .
Claim. For every , the following equations hold: for , where .
To prove the claim, we use the induction on . Since , then when the claim holds. Suppose that the claim is true for some , where . Multiplying (45) by , we obtain for . From (47), we derive for . Since is an -hyponormal operator, from (48) and Lemma 3 we obtain for , where . Combining (49) with the first equation of (47), we have for . Since is an -hyponormal operator, we obtain from Lemma 3 and (50) that for , where . From (49) and (51), we obtain for . Therefore, the proof of the claim is completed.
From the claim with , we have for . From Lemma 1 we derive that where denotes the orthogonal projection of onto . By applying the proof of (i), we obtain that is one-to-one and has closed range.

(iii)  Set . By (27), we have

Theorem 5. Let , where are mutually commuting, and both and are -hyponormal operators. If satisfy one of the conditions in Lemma 4, then is a subscalar operator of order , where is the appropriately chosen integer as in Lemma 4.

Proof. Let be a bounded disk in containing and consider the quotient space endowed with the Hilbert space norm, where , for (i), for (ii), and for (iii) in Lemma 4. The class of a vector or an operator on will be denoted by , , respectively. Let be the operator of multiplication by on . Then is a scalar operator of order and has a spectral distribution . Since is invariant under , can be well defined. Moreover, consider the spectral distribution defined by the following relation: for and , . Then the spectral distribution of commutes with , and so is still a scalar operator of order with as a spectral distribution. As in Lemma 4, if we define the map by then is one-to-one and has closed range. Since for all . In particular, is invariant under and is closed; it is a closed invariant subspace of the scalar operator . Since is similar to the restriction and is scalar of order , is a subscalar operator of order .

Corollary 6. Let , where are mutually commuting, both and are -hyponormal operators, and satisfy one of the conditions in Lemma 4. Then has property and the single-valued extension property.

Proof. From section one, we need only to prove that has property . Since property is transmitted from an operator to its restrictions to closed invariant subspaces, we are reduced by Theorem 5 to the case of a scalar operator. Since every scalar operator has property (see [6]), has property .

Define the quasi-nilpotent part of

Definition 7. An operator is said to belong to the class if there exists a natural number such that

Theorem 8 (see [13]). Every subscalar operator is .

Definition 9. An operator is said to be polaroid if every is a pole of the resolvent of .

Note that The condition of being polaroid may be characterized by means of the quasi-nilpotent part.

Theorem 10 (see [14]). An operator is polaroid if and only if there exists a natural number such that

Corollary 11. Every operator is polaroid.

Since a subscalar operator is , we have the following.

Corollary 12. Every subscalar operator is polaroid.

Corollary 13. Let , where are mutually commuting, both and are -hyponormal operators, and satisfy one of the conditions in Lemma 4. Then is polaroid.

If has SVEP, then and satisfy Browder’s theorem. A sufficient condition for an operator satisfying Browder’s theorem to satisfy Weyl’s theorem is that is polaroid. Then we have the following result.

Corollary 14. Let , where are mutually commuting, both and are -hyponormal operators, and satisfy one of the conditions in Lemma 4. Then Weyl’s theorem holds for and .

Observe that if has SVEP, then . Hence, if has SVEP and is polaroid, then satisfies -Weyl’s theorem [15, Theorem  3.10].

Corollary 15. Let , where are mutually commuting, both and are -hyponormal operators, and satisfy one of the conditions in Lemma 4. Then -Weyl’s theorem holds for .

Proof. Since is polaroid and has SVEP, then -Weyl’s theorem holds for .

In the following, is an analytic function on and is not constant on each connected component of the open set containing .

Corollary 16. Let , where are mutually commuting, both and are -hyponormal operators, and satisfy one of the conditions in Lemma 4. Then the following assertions hold:(i)Weyl’s theorem holds for ;(ii)-Weyl’s theorem holds for .

Proof. Since is polaroid and has SVEP, then satisfies Weyl’s theorem by [15, Theorem  3.14].
) Since is polaroid and has SVEP, then satisfies -Weyl’s theorem by [15, Theorem  3.12].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their cordial gratitude to the referee for his valuable advice and suggestion. This work was partially supported by the National Natural Science Foundation of China (11201126), the Natural Science Foundation of the Department of Education, Henan Province (no. 14B110008), and the Youth Science Foundation of Henan Normal University (no. 2013QK01).

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