`International Journal of Mathematics and Mathematical SciencesVolume 2012, Article ID 975745, 20 pageshttp://dx.doi.org/10.1155/2012/975745`
Research Article

On Some Normality-Like Properties and Bishop's Property (𝛽) for a Class of Operators on Hilbert Spaces

Mathematics Department, College of Science, Al Jouf University, Al Jouf 2014, Saudi Arabia

Received 11 December 2011; Accepted 19 February 2012

Copyright © 2012 Sid Ahmed Ould Ahmed Mahmoud. 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 prove some further properties of the operator (-power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator satisfying the translation invariant property is normal and that the operator is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator is subscalar of order ; that is, it is similar to the restriction of a scalar operator of order to an invariant subspace.

1. Introduction

Although normality of operators makes things easier, it rarely occurs and relaxing the normality condition is essential in the theory of operators on Hilbert spaces. One of the most important subclasses of the algebra of all bounded linear operators acting on a Hilbert space, the class of hyponormal operators, has been studied by many authors (see ). In recent years this class has been generalized, in some sense, to larger classes of the so-called -hyponormal, , posinormal, k-quasihyponormal classes, and so forth (see ).

In , Putinar showed that hyponormal operators are subscalar. This fact has led to far-reaching results, discovering deep properties of these operators. In this paper we extend that result to other generalized classes of operators.

Let be an infinite dimensional separable complex Hilbert space, let be a complex Hilbert space, and let be the algebra of all bounded linear operators from to . We write for . If , we will write and for the null space (also referred to as the reducing subspace) and the range of , respectively. The spectrum, the point spectrum, and the approximate point spectrum of an operator are denoted by , respectively. means the adjoint of .

An operator is(1)hyponormal if and only if ,(2)posinormal if and only if , or equivalently for some ,(3)-hyponormal if and only if ,(4)-quasihyponormal if and only if ,(5) if is invertible and ,(6)2-isometry if and only if (see ), where indicates the identity operator that is, if and only if An operator is said to be -power quasinormal (abbreviated as ), , if If , is called quasinormal. This class of operators being denoted by , that is, was studied by the author .

is called an -partial isometry if satisfies where is obtained formally from the binomial expansion of by understanding . The case when is called the partial isometries class. The class of -partial isometries was defined by Saddi and Sid Ahmed  who proved some properties of the class. See Proposition 5.4.

This paper is divided into five sections. Section 2 deals with some preliminary facts concerning function spaces. Section 3 includes our main results. There we study some properties of . In particular we show that an operator satisfying the translation invariant property is normal, and an invertible operator and its inverse have a common nontrivial invariant closed set provided that . Also we show that some of class satisfy an analogue of the single-valued extension for and have scalar extension. In Section 4, we give some results about the Berberian extension. In Section 5, we shall use some properties of the approximate spectrum to obtain some results on single-valued extension (SVEP) (see Section 2) property for the -partial isometries operators.

2. Spaces of Vector-Valued Functions

We will need the following function spaces.

Let be the coordinate in and let denote the planar Lebesgue measure. Let be a bounded open subset of . We will denote by the Hilbert space of square measurable (or summable) functions such that

Let denote the space of -valued functions analytic on , that is, . Equipped with the topology of uniform convergence on compact subsets of , is a Frechet space. Let denote the Bergman space for consisting of square measurable functions that are analytic on .

We denote by the orthogonal projection of onto .

Let us define now a special Sobolev type space. Let be a fixed nonnegative integer. The Sobolev space of order of vector-valued functions with respect to will be the space of those functions whose derivatives in the sense of distributions also belong to , that is,

Endowed with the norm becomes a Hilbert space contained continuously in ; that is, there is a constant such that .

Let for once and for all, whenever the definition is meaningful. We say that has the single-valued extension property at (abbreviated SVEP at ) if, for every open neighborhood of , the only analytic solution to the equation for all in is the constant function . We say that has SVEP if has a SVEP at every .

It is easily seen that an operator has SVEP if and only if, for each open .

The operator defined by is one to one.

Recall that, for a bounded operator on , the local resolvent set of at the point is defined as the union of all open subsets of such that there exists an analytic function which satisfies The local spectrum of at is the set defined by and obviously . It is clear from the definition that, has SVEP if and only if zero is the unique vector such that (see for more details ).

Recall that a bounded operator is said to have the Bishop’s property if for every open subset of the complex plane and every sequence of analytic functions with the property that uniformly on all compact subsets of , as locally uniformly on or equivalently, for every open subset of , the operator defined in (2.7) is one to one and has the closed range [11, Proposition ]. It is a very important notion in spectral theory. It is wellknown that every normal operator has Bishop’s property .

A bounded operator on is called scalar of order if it possesses a spectral distribution of order , that is, if there is a continuous unital morphism, such that , where stands for the identity function on and 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 the operator on such that for . This has a spectral distribution of order , defined by the functional calculus . Therefore is a scalar operator of order . Consider a bounded open disk which contains and the quotient space endowed with the Hilbert space norm. We denote the class containing a vector or an operator A on by or , respectively. Let be the operator of multiplication by on . As noted above, is a scalar of order and has a spectral distribution . Let . Since is invariant under every operator ; , we infer that is a scalar operator of order with spectral distribution . Consider the natural map defined by , for , where denotes the constant function identically equal to . In , Putinar showed that if is a hyponormal operator then is one to one and has closed range such that , and so is subscalar of order .

3. Further Properties of the Class [𝑛QN]

We start this section with some properties of -power quasinormal operators.

Theorem 3.1. The class has the following properties.(1)The class is closed under unitary equivalence and scalar multiplication.(2)If is of class and is a closed subspace of that reduces , then (the restriction of to is of class .

Proof. (1) Let be unitary equivalent to . Then there is a unitary operator such that which implies that . Noting that and inserting suitably, we deduce from (1.2) that and (1.2) follows for . Since for as the -th power or the adjoint, it follows that the left-hand side of (1.2) for reads which is , which is the right-hand side of (1.2). Thus, is of class .

Next we characterize a matrix on a 2-dimensional complex Hilbert space which is in . Since every matrix on a finite dimensional complex Hilbert space is unitarily equivalent to an upper triangular matrix and an -power quasinormal operator is unitarily invariant, it suffices to characterize an upper triangular matrix . From the direct calculation, we get the following characterization.

Proposition 3.2. For one has

We remark here that Proposition 3.2 offers the convenient criterion to find some examples of operators in . Also we observe that is not necessarily normal on a finite dimensional space.

Next couple of results show that does not have a translation invariant property.

Theorem 3.3 (see ). If and are of class , then is normal.

Theorem 3.4 (see ). If is of class such that is of class , then is normal.

It is natural to ask the following question: what is the operators in satisfying the translation invariant property? The answer to this question is provided by the following theorem.

Theorem 3.5. is of class for every if and only if is a normal operator.

Proof. Assume that is of class for every . Then (1.2) reads , which reduces on eliminating the common factor to By the binomial expansion, whence by arranging terms suitably, the extremal terms vanishing in view of (1.2),
Now note that from the second summand in (3.7), we may extract the extremal term and express it in terms of the remaining terms which contain to the power . Hence dividing (3.7) by and letting , we conclude that , whence the normality of follows.

Conversely it is known that normality is a translation invariant property; that is, if is normal, then is normal for every , and hence is of class .

The following proposition gives a characterization of an -power quasinormal operator.

Proposition 3.6. Let , , and . Then is of class if and only if commutes with .

Proof. Commutativity of and is equivalent to (1.2).

Proposition 3.7. Let be as in Proposition 3.6. Then if is of class , then commutes with and .

Proof. By (1.2),
In general, the two classes and are not the same (see ).

Proposition 3.8. If is both of class and , then it is of class , that is, .

Proof. By (1.2), so that may be transformed into .
It is known that if belongs to some , does not necessarily belong to the same class.

Theorem 3.9 (see ). If and are of class , then is normal.

Proposition 3.10. If an operator of class is a 2-isometry, then is of class for all integers .

Proof. From Proposition 3.8 it suffices to prove that is of class and of class because we may then proceed inductively.
Since is a 2-isometry, we have . Using , we may shift the power of to the left, arriving at ; that is, is of class .
In the same way, we may deduce that whence is of class .

Lemma 3.11 (see ). If is of class , then .

Proposition 3.12. If is both of class and such that is injective or is injective, then is quasinormal.

Proof. Since is of class , we have (3.9), which reads . If is injective, then so is and we have , whence is quasi-normal. If is injective, we may appeal to Lemma 3.11.

In the following theorem we prove some topological properties of the class .

Theorem 3.13. The class is an arcwise-connected, closed subset of equipped with the uniform operator (norm) topology.

Proof. By Theorem 3.1, (1), we see that the ray in through is contained in for every complex number , and therefore is arcwise-connected.
To see that is closed, we prove that any strong limit of a sequence in also belongs to ; that is, we let be a sequence of operators in converging to in norm: Hence it follows that whence converges strongly to .
Since the product of operators is sequentially continuous in the strong topology (see [12, page 62]), one concludes that converge strongly to . Similarly converges strongly to . Hence the limiting case of (1.2) shows that belongs to , completing the proof.

Proposition 3.14. If are of class , then both the direct sum and the tensor product are of class .

Proof. By the compatibility principle similar to (3.2), where indicates either the th power or adjoint, the proof follows.

A linear operator on is hypercyclic if there is a vector with dense orbit; that is, if there exists an such that orbit is dense in , and in this case is called a hypercyclic vector for .

An operator on is supercyclic if there exists a vector whose scaled orbit is dense; that is, if there exists an such that is dense in and in this case is called a supercyclic vector for .

Kitai  showed that hyponormal operators are not hypercyclic. We generalize Kitai’s theorem to the class .

Proposition 3.15. If is of class with , then is not hypercyclic.

Proof. If is hypercyclic, is hypercyclic, and hence by [13, corollary 2.4]. From Lemma 3.11 we have and, hence, , a contradiction.

Theorem 3.16 (see [14, Theorem 2]). If is not hypercyclic, then and have a common non-trivial invariant closed subset.

Proposition 3.17. If is of class and , then and have a common nontrivial closed invariant subset.

Proof . Since is of class and , it follows that is normal, and hence is normal. By [13, Corollary 4.5] and have no hypercyclic vector. Thus by , neither or has a hypercyclic vector. Therefore by  and have a common nontrivial closed invariant subset. Hence Theorem 3.16 completes the proof.

Proposition 3.18. Operators that are of class such that is not invertible are not supercyclic.

Proof. Assume that is of class and supercyclic. Considering the class being closed under multiplication by nonzero scalars, we may assume that . Since the supercyclic contraction satisfies property , is contained in the boundary of the unit disk [11, Proposition ]. Thus is invertible, and we have a contradiction.

Definition 3.19. An operator is algebraic if there is non-zero-polynomial such that .

The following proposition shows that some quasinilpotent -power quasi-normal operators are subscalar.

Proposition 3.20. If both and are of class such that is quasinilpotent, then is nilpotent, and hence is subscalar.

Proof. Since is quasinilpotent, . Hence by the spectral mapping theorem we get . Thus is quasinilpotent and normal. So is nilpotent, and is algebraic operator, and hence is subscalar.

Proposition 3.21 (see [7, Proposition 2.1]). For every bounded disk in , there is a constant such that for an arbitrary operator and we have where is the orthogonal projection of onto .

The next theorem is important for the proof of our main theorem, Theorem 3.27.

Theorem 3.22. Let be an arbitrary bounded disk in . If is of class and , then the operator is one to one.

Proof. Let such that , that is, Then for we have Hence for we get . Since is empty then is normal [9, Theorem 2.2]. Hence, Now we claim that Indeed, since is invertible for , (3.16) implies that Therefore
Since and , is invertible for , therefore; from (3.17) we have It is clear form (3.20); and (3.21) that Thus Proposition 3.21 and (3.21) imply where denotes the orthogonal projection of onto .
Hence . Since has SVEP, has SVEP. Also is analytic and for . Hence . Thus, is one to one.

Corollary 3.23. If and are of class with , for and . Then is one to one.

Proof. If is such that . Since , we get . Since is one to one, . Hence, . Since is one to one, .

The following corollary shows that the nilpotent perturbation of operators in satisfying SVEP satisfies SVEP.

Corollary 3.24. If an operator is a nilpotent perturbation of a 2-power quasi-normal operator , that is, , where is of class , and commute, and . If , then is one-to-one.

Proof. If is such that , then Hence for . We prove that for by indication. Since , . Since is one-to-one by Theorem 3.22. Assume it is true when , that is, . From (3.25), we get Since is one-to-one from Theorem 3.22, . By indication we have . Hence is one-to-one.

An operator is said to be the following.(1)It is left invertible if there is an operator such that , where denotes the identity operator. The operator is called a left inverse of .(2)It is right invertible if there is an operator such that . The operator is called a right inverse of (see ).

Corollary 3.25. If is of class with the property , and if is a left invertible operator with the left inverse , then the operator is one-to-one.

Proof. If is such that , then Hence for we have . From Theorem 3.22, we get for .
Thus, for . It follows that for . By application of [7, Proposition 2.1] with , we have where denotes the orthogonal projection of onto the Bergman space . Hence . From [17, Corollary 10.7], there exists a constant such that So . Thus, is one-to-one.

Corollary 3.26. If is of class with the property , and if is a right invertible operator with the right inverse then the operator , is one-to-one.

Now we are ready to prove our main theorem.

Theorem 3.27. If is of class with the property that and let be a bounded disk which contains , then the operator , defined by is one to one and has closed range, where is as in (2.11).

Proof. First, we will prove that if and are sequences such that then .
By the definition of the norm of a Sobolev space, (3.31) implies that From (3.32) we get
Since is normal, Since is invertible for , (3.32) implies that Therefore
Since and , it is clear that is invertible for . Therefore from (3.34), we have Hence, from (3.36) and (3.37) we get Then by Proposition 3.21, we have By (3.31) and (3.39), we have Let be a curve in surrounding . Then uniformly for by (3.40). Hence by Riesz-Dunford functional calculus But by Cauchy’s theorem Hence . Thus the map is one-to-one and has closed range.

Corollary 3.28. If is of class with the property that , then T is subscalar of order .

Proof. Consider an arbitrary bounded open disk in that contains and the quotient space given in (2.11).
Let be the multiplication operator by on . Then is a scalar operator of order , and its spectral distribution is where is the multiplication operator by . Let . Since is invariant under every operator , we infer that is a scalar operator of order with spectral distribution .
Let be the operator from into . Then we have the following commutative diagram (3.46) By the previous theorem the operator is a topological isomorphism of into . The relation shows that is -invariant. Hence is an extension of the operator , so this operator is subscalar. Since is invertible on , then the operator is subscalar of order . On the other hand from [18, Theorem 4.3] we deduce that and the theorem is proved.

Corollary 3.29. If is of class with the property that , then has Bishop’s property .

Proof. It follows from Corollary 3.28 and [18, Lemma 2.1].

In  the authors study some operators with the single-valued extension property. In the following propositions we extend some of these results to operators with the Bishop’s property .

Proposition 3.30. Let be the following triangular operator matrix Assume that is of class and satisfies for and is nilpotent. Then has Bishop’s property .

Proof. Let be a sequence of analytic functions such that uniformly on every compact subset of an open set of ; then we have Since , and hence if . Since from (3.48) . By the same reason, . By repeating this procedure, we finally achieve uniformly on . Then we obtain the following equation: uniformly on every compact . Since has Bishop’s property from Corollary 3.29, uniformly on . By repeating this process we prove that uniformly on .
Hence converge uniformly to 0 on any compact subset of , and so has the Bishop’s property .

Proposition 3.31. Let be as in Proposition 3.30. Then if has Bishop’s property for , then has Bishop’s property .

Proof. The proof is identical to the proof of Proposition 3.30.

Proposition 3.32. Let be the following triangular operator matrix: Assume that is of class and . If for and , then has Bishop’s property for .

Proof. Let be a sequence of analytic functions such that uniformly on every compact subset of , then we have for for . Since has Bishop’s property , we get that uniformly on for . We have and for , Since has Bishop’s property , uniformly on for . Thus, has Bishop’s property .

Proposition 3.33. Let be as in Proposition 3.31. Then if has Bishop’s property and for and , then has Bishop's property for .

Proof. The proof is identical to the proof of Proposition 3.32.

4. Berberian Extension

Denote by the space of all sequences , with such that is bounded. Let denote the subspace of all null sequences of (those such that ). If we set for every sequence , this defines a seminorm on , which is zero exactly on the elements of . By means of the space and the Banach limits, Berberian  constructed an extension of and obtained a homomorphism form operators to operators such that is an extension of .

Theorem 4.1 (Berberian extention ). Let be a complex Hilbert space. Then there exists a Hilbert space and a map satisfying: which is an *-isometric isomorphism preserving the order such that(1), (2), (3), (4), (5), (6), (7), (8)if is a positive operator, then .

An operator is said to be reducible if it has a nontrivial reducing subspace. If an operator is not reducible, then it is called irreducible.

Proposition 4.2 (see ). If is an irreducible operator, then is an irreducible operator.

Lemma 4.3. Let be a subset of , , such that , let be a sequence of analytic functions, and let the Taylor expansion of be If is uniformly bounded on i.e.,, then

Proof. For all and with , by Cauchy’s integral formula, we get the following inequality:

Remark 4.4. Let be an open subset of . A sequence of analytic functions converges uniformly to 0 on every compact subset of if and only if for any and any there exists and such that and for all .

5. Single-valued Extension Property for 𝑚-Partial Isometries

In this section we examine the properties of SVEP and Bishop’s property for some -partial isometries operators by using an approach which is different from that used in Section 3. We recall the definition of an -partial isometry given by (1.4) and the operator .

Definition 5.1 (see ). An operator is called an -isometry if

Remark 5.2. It is easy to see that is an -partial isometry if and only if which shows that the class of -partial isometries generalizes those of -isometries and partial isometries.

Theorem 5.3 (see ). If is reducible, that is, if it has a nontrivial reducing subspace , then the following properties are equivalent.(1) is an -partial isometry.(2) is an -isometry.

Proposition 5.4 (see ). Let be a reducible -partial isometry. Then(1) implies , that is, if for some sequence of bounded vectors , then ,(2) implies ,(3)eigenvectors of corresponding to distinct eigenvalues are orthogonal, that is, if .

Lemma 5.5. Let be a reducible -partial isometry and let and , be sequences of bounded vectors in such that and Then we have

Proof. We may assume that . Then from Proposition 5.4(1) we have as . Hence, which implies (5.4) in view of and the proof is complete.

Theorem 5.6. Any reducible -partial isometry has SVEP.

Proof. Let be a bounded subset of and let be an analytic function such that Since (Proposition 5.4(3)), we have This shows that .

Lemma 5.7. If is an -partial isometry, then is also an -partial isometry.

Proof. It is a consequence of the properties of (see Theorem 4.1).

Theorem 5.8. If is an -partial isometry with a nontrivial reducing space , then has the single-valued extension property (SVEP).

Proof. To prove that has SVEP, let . Since by Theorem 4.1, . In particular if , then . Hence, for and with . In a similar way as in the proof of Theorem 5.6, we can see that has SVEP.

Acknowledgment

The author would like to express their gratitude to the referees for their helpful and many valuable suggestions.

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