Similarity of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.1996pt" id="M1" height="15.599pt" version="1.1" viewBox="-0.0657574 -11.3994 18.0806 15.599" width="18.0806pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g><g transform="matrix(.012,0,0,-0.012,11.377,4.134)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg>:</span> Operators and the Hyperinvariant Subspace Problem

Segres, Abdelkader; Bachir, Ahmed; Ould Ahmed Mahmoud, Sid Ahmed

doi:https://doi.org/10.1155/2024/9943902

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 9943902 | https://doi.org/10.1155/2024/9943902

Similarity of : Operators and the Hyperinvariant Subspace Problem

Abdelkader Segres,¹Ahmed Bachir,²and Sid Ahmed Ould Ahmed Mahmoud³

Academic Editor: G. Muhiuddin

Received02 Jan 2023

Revised28 Feb 2024

Accepted04 Mar 2024

Published17 Apr 2024

Abstract

In the present paper, we first show that the existence of the solutions of the operator equation is related to the similarity of operators of class , and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of for some singular inner functions . As an application, we prove that every -quasinormal operator and -centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.

1. Introduction

Let be a separable infinite dimensional complex Hilbert space and let be the algebra of all bounded linear operators acting on (see [1] for basics and fundamentals). The commutant of , denoted by , is the algebra of all operators such that . A closed subspace is called a nontrivial hyperinvariant subspace for if and for every . In particular, if , then the subspace is called a nontrivial invariant subspace for . The invariant subspace problem asks whether every operator has a nontrivial invariant subspace with . In a similar fashion, the hyperinvariant subspace problem asks whether every bounded linear operator such that has a nontrivial hyperinvariant subspace. These problems are still unresolved, especially for operators such that for every nonzero in .

A power-bounded operator of class which commutes with a nonzero quasinilpotent operator has a nontrivial invariant subspace. In the hyponormality case, it is well known that in [2], Kubrusly and Levan have shown that if the strong limit , defined below, is a projection for every biquasitriangular contraction , then every contraction not in has a nontrivial invariant subspace. We recall the following standard definitions: for , is a normal operator if , is a quasinormal operator if , is subnormal if there exist a complex Hilbert space and a normal operator such that is an invariant subspace for and is the restriction of to (i.e., ). is hyponormal if . The proper inclusions are well known (see, e.g., [3]).

Recall that a unitary operator is singular (resp. absolutely continuous) if its spectral measure is singular (resp. absolutely continuous) with respect to the Lebesgue measure on the unit circle. Any contraction can be decomposed uniquely as the direct sum , where and are singular and absolutely continuous unitary operators, respectively, and is a completely nonunitary contraction. is said to be absolutely continuous if in this decomposition is absent. For this type of decomposition for polynomially bounded operators, see [2, 4].

It is well known that the equation , where is the unilateral shift of multiplicity one, characterizes the class of Toeplitz operators, and that this type of equations for contraction operators was studied by many authors as in [5] and the references therein. For the invariant subspace problem, it is known that it was solved for the class of subnormal operators but the hyperinvariant subspace problem is still open for this class (for certain partial results, readers may wish to look at [6]). In [7], it was shown that if is a hyponormal operator with a thin spectrum, then it has a nontrivial invariant subspace, and in the case, some partial results were obtained by Kubrusly and Levan in [8], who have proved that if a hyponormal operator has no nontrivial invariant subspace, then is either a proper contraction of class or a nonstrict proper contraction of class for which the strong limit of is a completely nonprojective nonstrict proper contraction. For more details, see [4, 9–16].

In the present paper, we show that the existence of nontrivial solutions of the equation , where is the unilateral shift of multiplicity one on the hardy space and is a polynomially bounded operator on a Hilbert space and is related to the similarity of operators of class . In other words, has nontrivial solutions if and only if the operator matrixis similar to .

Then, we study the hyperinvariant subspaces problem for polynomially bounded operators of class , and we give sufficient conditions for the existence of nontrivial hyperinvariant subspaces for hyponormal operators of class .

Now, we summarize our main results.

Let be a polynomially bounded absolutely continuous operator. Iffor some nonzero such that the operator matrixis similar to . Then, has nontrivial hyperinvariant subspaces.

The nontrivial hyperinvariant subspaces obtained are the closure of where is a singular inner function. The operator is the function of obtained using the -functional calculus defined for absolutely continuous polynomially bounded operators [2, 4]. As an application, we prove that if is a -quasinormal operator, then has a nontrivial hyperinvariant subspace. In particular if is a completely nonnormal quasinormal operator, then the nontrivial hyperinvariant subspaces of are the closure of where is a singular inner function. In the case where is a centered operator, we give a refinement of the result by showing that we may take .

Next, we provide other conditions sufficient for the existence of nontrivial hyperinvariant subspaces for .

2. Preliminaries

2.1. Notations and Definitions

Throughout this paper, denotes an infinite dimensional complex separable Hilbert space with inner product and denotes the space of all bounded linear operators acting from to . The kernel and the range of an operator will be denoted by and , respectively, and the rank one operator ; is defined by , for all . The closure of a subspace of will be denoted by . For a contraction , the operators and are the defect operators and is the commutator of .

Let be bounded linear operators on the Hilbert spaces and , respectively. Consider the set

If there is an operator with a dense range, we set . An operator will be said to be a quasiaffinity if it is injective and has a dense range and the operator is a quasiaffine transform of the operator and if there exists a quasiaffinity , we set .

2.2. Strong Limit for Contraction Operator

If is a contraction, then is a nonincreasing sequence of nonnegative contractions so that it converges strongly to an operator which satisfies the following properties: , as for all , for all and there exists an isometry on such that

Furthermore, the subspace is a hyperinvariant subspace. We say that is of class , that is strongly stable, if and is of class if . is of class if is of class and is of class if . For more details, see [4, 14].

We denote by the open unit disc and by the unit circle. Let denote the normalized Lebesgue measure on the unit circle (i.e., ) and let denote the space of all complex-valued Lebesgue measurable functions on such that is finite. As such, is a Hilbert space, a simple calculation using the fact that shows that this space has a canonical orthonormal basis given by , for all ; being the set of integers and denotes the identity function, i.e., ; and in the sequel, we set .

The Hardy space is the closed linear span of . The operators of multiplication by the identity function on the spaces and are the unilateral forward shift in defined by and the unilateral forward shift in defined by . It is clear that the bilateral forward shift on has the following form with respect to the decomposition :

For a Borel set , we write and the operator of multiplication by the identity function on the space will be denoted by .

Definition 1 (See [11]). A dissymmetric weight is a nonincreasing, unbounded function satisfying the following conditions:(1)(2)(3) when

Definition 2. (1)An inner function is a bounded analytic function on such that for almost every in , where is the radial limit of (i.e., ).(2)Let be a positive, finite singular (with respect to the Lebesgue measure ) Borel measure on . A singular inner function is an analytic function defined by If denotes the point mass at , then This type of inner function is called an (singular) atomic inner function.(3)An outer function is an analytic function on of the form where is a real constant and is a real-valued function in .

Remark 3. It is well known that the only nonconstant invertible inner functions in the Hardy spaces are the outer functions. For more details, see [17, 18].

Theorem 4 (See [11]). Let be a dissymmetric weight. Then, there is a singular inner function such that and

Lemma 5 (See [19]). Let be a sequence of positive numbers such that . Then, there exists a dissymmetric weight such that for sufficiently large .

Definition 6. An operator is said to be polynomially bounded if there exists such that for every polynomial , where .

We denote by for the set of polynomially bounded operator in . It is well known, by von Neumann’s inequality, that every contraction operator is polynomially bounded.

Proposition 7 (See [19]). Let be an absolutely continuous operator and let be a singular inner function. Iffor some , then

Lemma 8 (See [7]). Let . If is a polynomially bounded operator, then there is a contraction operator such that . Conversely, if is a contraction operator, then there is a polynomially bounded operator such that .

3. Similarity of Operators

Let be an operator on defined for every by

Set

Following [20], and are called the subspaces of cocycles and coboundaries, respectively.

Proposition 9 (See [20]). (1) if and only if , for every .(2) if and only if the operator matrix is similar to .

It is clear that for every and are hyperinvariant subspaces (not necessarily closed) for .

Remark 10. We note here that if is a unitary operator on , then . If is a unilateral shift on , then (see [20, 21] for further details).

Let be an operator (not necessarily bounded) from to defined by

It is easy to check that for every bounded operator on .

Lemma 11. Let . Then, is a bounded operator from to for all , where is a quasiaffinity in and is a contraction operator.

Proof. According to Lemma 8, there exists a contraction operator and a quasiaffinity in . Let . Then, for all , we haveA simple calculation shows, for all , thatHence,where is the strong limit defined in Section 2 for the contraction . Hence, an easy computation shows that the following operatoris bounded for every .

Proposition 12. Let . Then, the following conditions are equivalent:(1)(2)The equation has nontrivial solutions in (3)There exist nonzero operators such that

Proof. : If , then there is such that ; . Hence, , and then .
Conversely, let be a nonzero solution of the equation , then . So, we either have (i) or (ii) .
(i). If that is , then by Remark 10, there exist and such that . Thus, . If , then is a nontrivial invariant subspace for . Hence, by Beurling’s theorem, there exists an inner function such that and the restriction is a unilateral shift on . Thus, . So, using the same argument as in (i), we get . (ii). If , then for every , there is a scalar such that . The function is a bounded functional. Hence, by Riesz representation’s theorem, there exists such that for every . Therefore, for every . This means that .
: If , then there is such that , . Setwhere is an operator from to and .
By the same argument as in Lemma 11, it is seen that is a bounded operator if , and therefore, is a bounded operator.
An easy computation then shows that , and as the converse is clear, the proof is complete.
In what follows we show that if is a polynomially bounded operator of class , then .

Corollary 13. If is a polynomially bounded operator of class , then , and if is a contraction operator, then .

Proof. According to Lemma 8, we can suppose that is a contraction operator. Then, by Lemma 11, we can find a certain (in the range of ) such that the operator is a bounded operator
Since , by Subsection 2.2, we getSince the strong limit is an injective positive operator , . Hence, the equation in has a nontrivial solution.The result now follows from Proposition 12. It follows from the proof of the previous proposition that the solutions of the equation in have the form , where . If , then there is such that . An easy computation shows that for some .
Now, we recall some well-known facts: an operator is said to be binormal if and commute, see [5, 20]. An operator is said to be centered if the following sequenceis commutative. In [4], Morrel and Muhly showed some properties and obtained a nice structure of centered operators. We also recall that binormal operators are called weakly centered operators in [18]. The following result is due to V. Paulsen, C. Pearcy, and S. Petrovic [18].

Theorem 14. Every power bounded centered operator is similar to a contraction.

It is easy to see that the following results hold true.

Lemma 15. The class of binormal is self-adjoint and closed under multiplication by complex numbers, taking inverses and formation of direct sums.

Proposition 16. If is a centered contraction operator, then the asymptotic limit commutes with .

Proof. Since is the strong limit of the sequence , ,Since is a centered operator, commutes with for all , and therefore,Hence,This means that , for all . Accordingly, , as needed.
As a consequence of Corollary 13 as well as the preceding proposition, we obtain

Corollary 17. If is a centered contraction operator of class , then

4. Hyperinvariant Subspaces

First, we recall some well-known facts in complex analysis: for every analytic function in the function defined on by is analytic in and .

If is an absolutely continuous -operator and , then , see [2]. For and for , we set

, for every .

Then,

If is a singular inner function, then it has no zeros in , and so the function is analytic in , i.e., .

From the equality , we get

Theorem 18. Let be an absolutely continuous operator. Iffor some nonzero , then has nontrivial hyperinvariant subspaces.

Proof. Let . Then, by Proposition 12, there existsin , where is an operator from to and . Since , may be written as .
Set . If , thenIt then follows from , by an easy computation, that .
Similarly, we havewhere .
On the other hand, we haveBy Proposition 7,so thatSuppose to the contrary that, for every , has no nonzero eigenvectors, that is,Then,Hence, by relations (22), (41), (42), and (43), we getIn particular,Since , it is seen thatTherefore,by (40), we have , and therefore, is an eigenvalue for .
So, if the point spectrum of is empty, then there exists such thatThus, by (44), are nontrivial hyperinvariant subspaces for .

Corollary 19. Let . If there exists a solution to the equation such thatthen either the point spectrum of is not empty or has nontrivial hyperinvariant subspaces of the form , where is a singular inner function.

Remark 20. Since the operatoris of class , by Proposition 9 and Corollary 13, there exists a dense linear manifold for (otherwise it will be a nontrivial hyperinvariant subspace for ) such that the operator is similar to . Hence, Theorem 18 means that if is an absolutely continuous operator and there exists some such that is similar to , andthen either the point spectrum of is not empty or has nontrivial hyperinvariant subspaces of the form , where is a singular inner function.

In the sequel, we give some applications of the previous theorem.

Proposition 21. If is a contraction quasinormal operator, then for every nonzero , there is an increasing sequence of positive numbers : and such that

Proof. Let ; . Recall that if is a quasinormal operator, then . Hence,By the hyponormality of , we get that for every . Since is of class , we claim that there is a singular inner function such thatIndeed, by Lemma 5, there exists a dissymmetric weight such that for sufficiently large . By Theorem 4, there exists a singular inner function such that. For sufficiently large , we getThat means that , for every .
Next, let be a sequence defined by ; . It is clear that is a positive increasing sequence. By Remark 3, . That is, is an unbounded sequence . An easy computation shows thatfor every .
Therefore, , for every .
Thus, by (45), we get .

Lemma 22. If is a contraction quasinormal operator, then .

Proof. If , then there exists such thatMultiplying (52) by and using the quasinormality of give.In other words, , for each .

Theorem 23. Let be a -quasinormal operator, then either the point spectrum of is nonempty or has nontrivial hyperinvariant subspaces of the form , where is a singular inner function.

Proof. According to Theorem 18 and the previous lemma, it suffices to show thatSet . By Proposition 21, there exists a positive unbounded sequence such that (47) holds.
By Lemma 5, there exists a dissymmetric weight such thatfor sufficiently large . Also, by Theorem 4, there exists a singular inner function satisfying (10). Therefore,Finally, the result follows from Theorem 18, and this completes the proof.
As a consequence of Theorem 18 and Corollary 17, we get the following result that gives a refinement of the condition cited in Theorem 18 for the centered -operators.

Theorem 24. Let be a -centered operator. If there exists such thatfor some singular inner function , then has nontrivial hyperinvariant subspaces of the form .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the anonymous referees for their helpful comments that improved the quality of the manuscript. The authors extend their appreciations to the Deanship of Scientific Research at King Khalid University for funding this work through Small Research Project grant number (G.P.R.1/151/43).

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Copyright © 2024 Abdelkader Segres et al. 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.

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