In a separable Hilbert space, the most important bases are orthonormal bases. Second in importance are those bases that are bases equivalent to some orthonormal basis. They will be called Riesz bases, and they constitute the largest and most tractable class of bases known. For a linear system, the property that the eigenvectors of the linear system forms a Riesz basis for the state Hilbert space is one of the most important features from both theoretical and practical points of view.
Mathematically, the general Riesz basis property is developed in the context of nonharmonic Fourier series, which originated from the works of Paley and Winer and was developed later by many former Soviet mathematicians; we can quote most of them [16, 8, 15, 9].
In this paper, we consider on a Hilbert space the linear system
where is an unbounded linear operator on and .
If the operator generates a -semigroup on , then the solution of (1.1) can be written in as .
In this case, we will also claim that this property be essential in our paper. In many applications, the differential operator arises from a partial differential equation, for which it is know that (1.1) has a solution. Hence the assumption that generates a -semigroup is not strong. However, for our result, the semigroup property does not suffice; we need that generates a group, i.e., (1.1) possesses a unique solution forward and backward in time. Since we also assume that the eigenvalues lie in a strip parallel to the imaginary axis, the group condition is not very restrictive. In order to apply the Pavlov theorem  to get a Riesz basis of expansion of eigenvalues of the general linear system in Hilbert space, we need to relate eigenvectors of the system operator with the exponential system through its spectrum.
In spectral theory, to show that system (1.1) satisfies the spectrum-determined growth condition is an interesting but difficult problem. The spectrum-determined growth holds for the -semigroup generated by . However, verifying this condition is difficult because the semigroup usually may not have a workable expansion. So using the Riesz basis of the eigenvectors of the studied system becomes an attractive alternative.
In this model that we study in this paper, we need to discuss properties of the -semigroup. Since, very frequently, we merely have information about the infinitesimal generator , it is important to know how to deduce these properties from those of . It is long this line of thoughts that we try to use spectral information of to investigate the spectrum-determined growth. The approach that we take to prove our result is different to the one taken in [4, 6, 8, 10, 11].
Since this Riesz basis property is so important, there is extensive literature on this problem; we refer to the works [4, 6, 9, 10] where the same problem is treated for a specific perturbation of a closed linear operator. The basic approach is to estimate eigenvalues, the exponential family of eigenvalues, and then find some transformation to transform the set of the family of a perturbed operator to an orthonormal basis of eigenvectors.
We concentrate our attention in the second part of this paper to give a unified treatment for the following general class of linear hyperbolic systems:
with a strictly positive function , which was first studied by Intissar  in the case where the function is considered constant, equal to one, with a unique restrictive termination. A nice summary of the later development can be found in  under some transformation on the operator . Among them, a powerful concept of a Riesz basis was introduced to prove that the eigenvectors of form a Riesz basis in Hilbert space. However, it appears that a mathematical treatment of this problem has not yet been undertaken in the case where the operator is given in the above form (1.3). The exact question is to know whether the eigenvectors of this operator (1.3) form a Riesz basis in Hilbert space. Answering this equation is important in this model. It is the topic that we shall study in this paper.
The main concerns of this paper are
Riesz basis property of exponential family,
expansion of the solution in terms of the eigenvectors under nonharmonic Fourier series,
spectrum-determined growth condition.
We show that the spectrum of system (1.1) consists of zeros of a sine-type function where the expansion family and its eigenvectors form a Riesz basis.
Now let us outline the content of this paper. In Section 2, we recall some preliminary results and definitions that we will be using. In Section 3, we introduce our basic result. Section 4 is devoted to applying the main result for the linear hyperbolic system. In other words, we show that the Riesz basis property as well as the spectrum-determined growth condition hold for the studied system.
2 Preliminary results
For the sake of completeness, we first recall the following notions and preliminary results which will be used in the sequel.
Let be a Hilbert space, and let T be a closed operator with dense domain . Let (resp. ) denote the spectrum (resp. the resolvent set) of T. For any , is called the resolvent of T.
In the remaining part of this section, we will give the characterization of the Riesz basis of an exponential family developed by Pavlov . Before stating this result, we shall first recall some basic facts about functions of sine type due to Avdonin and Ivanov  formulated as follows:
An entire function is said to be of exponential type if the inequality holds for some positive constants A and B and all complex values of z. The smallest of constants B is said to be the exponential type of .
An entire function f of exponential type is said to be of sine type if
the zeros of f lie in a strip for some ,
there exist , and positive constants such that for all .
We want to mention Pavlov’s characterization of exponential Riesz bases .
Theorem 2.1 ().
Let be a countable set of complex numbers. The family forms a Riesz basis in if and only if the following conditions are satisfied:
the sequence lies in a strip parallel to the real axis, ;
is separated in the sense that ;
the generating function of the family on the interval satisfies the Muckenhoupt condition
where is the set of all intervals of the real axis.
It is seen that if the generation function f of is of sine type, then conditions (i) and (iii) of Theorem 2.1 are always satisfied.
Before recalling the results of the next theorem that are used subsequently, we begin by recalling the useful definition due to M. V. Keldys.
Let K be a compact operator in a Hilbert space . Then K is said to belong to the Carleman class , , with order p if the series converges, where the are the eigenvalues of the operator .
In the particular case , is exactly the space of Hilbert–Schmidt operators, and for , is called the space of nuclear operators or the space of trace class operators on .
Now let T be a closed operator with dense domain in a Hilbert space . We suppose that , and we put . If K belongs to the Carleman class for some , then Lidskii  obtained estimates for the growth of the entire functions and as . For the operators of this class, Lidskii  proved the following result.
If K belongs to the Carleman class for some , then can be represented as the ratio of two entire functions for which the inequalities and are valid for all ( and being positive constants).
In this work, we are interested to proof the Riesz basis property of the eigenvectors associated to unbounded linear operator involving the concept of the exponential Riesz basis family. For this purpose, we need to review the following theorem due to Aimar, Intissar and Paoli [1, 2].
Let T be a closed linear operator with dense domain on a Hilbert space . Suppose that
there exists such that is a compact operator,
there exists such that as ,
for sufficiently large, the resolvent is bounded on each m rays centered at the origin, divided on the all complex plane such that
( , where designates the entire part of ),
Then the system of generalized eigenvectors of T is complete in .
3 Main results
In this section, we show that the eigenvectors of system (1.1) form a Riesz basis in Hilbert space , which makes us conclude its spectrum-determined growth.
Let be a linear unbounded densely defined operator on a separable Hilbert space satisfying that
generates a -semigroup,
the resolvent of is compact,
the eigenvalues are simple,
the family is a Riesz basis for , ,
the system of eigenvectors of is complete in .
Then the eigenvectors associated to the operator form a Riesz basis in .
For the convenience of the reader, we give here a short proof of this theorem which follows as a corollary of the results given by Xu and Yung .
We define the -invariant spectral subspace of by , where we define the set as
where denotes the Riesz projection on associated to .
Since the system of the generalized eigenvectors of is complete in , then, for any and , the function is in the subspace , where we define by the span generated by the family as
Hence, for any , , and we can write them as
Moreover, we infer from the Riesz property of the exponential family in that there exist two positive constants and such that
According to assumption (i) with the property of the semigroup that for and (see [14, Theorem 2.2, p. 4]), we get
Since is the generator of a -semigroup in a Hilbert space , then the adjoint of has the same property [7, p. 2354]; in other terms, is the generator of a semigroup with the same conjugate eigenvalues as , and each is an isolated eigenvalue of with
For any , and
Reasoning in a similar way as above, we can easily check that
For any , we can write them as
So a short computation reveals that
Taking the limit, we get
Hence we deduce that
This makes us conclude that the series converges in with the sum equal to f and yields that forms a Riesz basis in . ∎
As a consequence of the foregoing result from the Riesz basis property of eigenvectors, we have the following result.
System (1.1) satisfies the spectrum-determined growth assumption, i.e., , where
To illustrate the applicability of our main result, we concerns our attention in the next section to present an application of Theorem 3.1 to prove the Riesz basis property of the linear hyperbolic system.
4 Application to a linear hyperbolic system
In this section, we shall show that system (1.2) has a set of eigenvectors which form a Riesz basis in Hilbert space . The key idea is to apply some basic result of Pavlov , which provides a useful way to verify the Riesz basis property for the eigenvectors of a linear operator in Hilbert space.
Since the matrix is symmetric with distinct eigenvalues , , introducing the change of variables
our system becomes
with and defined by
In this part, our goal is to give some important properties about a linear hyperbolic system to verify the assumptions of Theorem 3.1 in several steps. We shall assume without loss of generality that , , and . In the following, we establish some preliminary results about the operator .
is dense on .
The canonical injection i from into is compact.
(i) It is seen that is a closed subset of the Sobolev space which is dense in . Then assertion (i) follows immediately.
(ii) On the other hand, we infer from the Sobolev theorem [5, Theorem VIII.7, p. 129] that the injection from into is compact. So the injection from into is too. ∎
The following statements hold:
The operator generates a -semigroup on .
The resolvent expression of can be represented as
(i) Since the entries of the matrix are real-valued -functions in x, generates an evolution operator.
(ii) First, to get the resolvent expression of the operator , we consider the inhomogeneous equation expressed as
where , and is arbitrary in .
An elementary calculation by the variation of constants reveals that the general solution of equation (4.4) is given by
where we designate by the matrix
As a preparation for attacking the expression of , it is sufficient to solve by recalling the resolvent of . So we get
Denote by the expression
which is an entire function on λ. Using the above equality with this notation, we get
A short computation shows that a complex number λ belongs to the resolvent set of if and only if ; in this case,
which completes the proof. ∎
is a discrete operator in ; in other words, for any , is compact on . Therefore, the spectrum consists only of isolated eigenvalues.
It follows from Proposition 4.1 together with the fact , that has a compact resolvent on . Thus its spectrum consists only of isolated eigenvalues, which are exactly the zeros set of . ∎
For each , all eigenvectors associated with λ can be represented as
The next theorem gives information about the distribution of the eigenvalues of the operator .
Each eigenvalue of is simple and given by
An elementary calculation reveals that the zeros set of consists of an infinite number of complex eigenvalues of the operator , which are formulated by the form
where are defined as above and is defined by
Obviously, we observe from the above formula that each eigenvalue is simple. ∎
The corresponding eigenvectors associated to are
Therefore, the eigenvectors associated to have the asymptotic form
It is easily seen that
is invertible in x and .
The motivation for our studies is based on proving the Riesz basis property of in . To obtain such property, explicit estimates for sine-type functions turn out to be good.
forms a Riesz basis for .
It should be noted that our results are aimed to provide a description for the Riesz basis property of since the real part of the eigenvalues is independent of n.
To claim this, let be defined by . The zeros set of is related by
Obviously, is uniformly bounded on the real axis, and hence g belongs to the Cartwright class. Thus its indicator diagram is an interval. Furthermore, it is easy to show that is of exponential type with type . In fact, let . Then
Moreover, from the last expression, we have
where , and all whenever y is sufficiently large. In what follows, item (ii) of Definition 2.2 is satisfied.
Moreover, it is seen that the zeros are separated. Indeed, let . Then we have
As a consequence of the above sentences, we get the Riesz basis property by Theorem 2.1 for in and hence for in . ∎
Let be the generator of a -semigroup in a Hilbert space . Then the spectrum of its adjoint is given by .
Since is a Hilbert space and is the generator of a -semigroup , then is also a -semigroup, and its generator is . Moreover, its spectrum consists of an isolated eigenvalue , where . ∎
The resolvent of the operator belongs to the Carleman class for every .
A short computation from the expression of the eigenvalues of the operator , which is given in Theorem 4.3, shows that
Obviously, , so let , and let be the eigenvalue of the operator . It is clear that
Then belongs to the Carleman class for every . ∎
As a consequence of Theorem 2.2, we have the following corollary.
The resolvent of can be represented as two entire functions
Let . Then we have
Denoting , then
Now we are in the position to provide the denseness of the eigenvectors of the operator .
The system of the eigenvectors of the operator is dense in .
From Theorem 4.4, the resolvent of belongs to the Carleman class for since is an entire function of λ and the orders of both entire functions of and are less than or equal to 1. That is, there is a such that
As generates an analytic semigroup, it follows that is uniformly bounded in both real and imaginary axis. We may assume without loss of generality that is uniformly bounded in the right complex plane, particularly on the imaginary axis.
By assumption, is bounded on the boundary , and for all , where is chosen so that . Applying the Phragmen–Lindelof theorem to in each (see [17, Theorem 10, p. 80]), we known that is uniformly bounded in and so in the whole complex plane. Therefore, all conditions of Theorem 2.3 are satisfied with for , and . This result in the completeness of the generalized eigenvectors of on . ∎
We summarize all the results proved in this section by mentioning the following result.
The eigenvectors associated to the operator form a Riesz basis in .
From the previous results presented in this part, we infer that all the hypotheses of Theorem 3.1 are fulfilled, which makes us conclude that the system of eigenvectors associated to forms a Riesz basis in the state Hilbert space . ∎
As a consequence of the spectral mapping theorem for the -semigroup generated by , we have the following result.
Our system satisfies the spectrum-determined growth assumption, i.e., .
All the results in this paper cover the main results of [8, 11] as a special case and remove some additional conditions imposed there. The exponential family of the eigenvalues is then readily established from an expression of the eigenvalues, which is also obtained in the process of verification of the Riesz basis property.
Our work represents a specific case of the results given in [8, 11]. Our new contribution in this paper is considering a perturbed operator matrix in terms of the variable x. New results and techniques are given to reach the Riesz basis property of the eigenvectors associated to an unbounded operator matrix . However, this kind of study makes us conclude its spectrum-determined growth. More precisely, the study involves an elegant use of the notion of Pavlov’s characterization of the exponential Riesz basis properties in order to deduce the Riesz basis property for the eigenvectors associated to the operator . The abstract result is illustrated by an example of a linear hyperbolic system.
The authors would like to thank the referees for their valuable suggestions for the revision of the paper and for attracting the authors’ attention to .
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Published Online: 2019-05-29
Published in Print: 2019-06-01