Research Article | Open Access
Norm Estimates for Solutions of Polynomial Operator Equations
We consider the equations and , where , , are given linear bounded operators in a Banach space and is to be found. Representations of solutions are derived. In the cases of Euclidean and Hilbert spaces, norm estimates for the solutions are suggested.
1. Introduction and Statement of the Main Result
The objects of this paper are the equationswhere , , , , are given linear bounded operators in a Banach space with a norm and is to be found.
There is a long tradition of finding different expressions for the solution of an operator equation in the form of operator integrals and series, some prominent examples of which occur in the works of E. Heinz, M. G. Krein, M. Rosenblum and R. Bhatia, C. Davis, and A. McIntosh. A comprehensive summary of these is contained in  and . This tradition was continued in , which deals with the equationClearly, (3) is a particular case of (2).
The special case of (2) is the much studied Sylvester equation, of great interest in operator theory, numerical analysis, and engineering; compare [4–6] and references therein. Let be the operator adjoint to . The equationis the famous Lyapunov equation playing an essential role in the stability theory of differential equations. The equationis an example of (1); in particular, the discrete Lyapunov equationis an important tool in the theory of difference equations.
About other operator equations see [7–9] and references therein. In particular,  deals with necessary and sufficient conditions for the existence of solution to the systems of the general solution to a system of adjointable operator equations over Hilbert modules. In  the nonlinear operator equations of the form and are considered. It should be noted that in the finite dimensional case the numerical methods are well developed. Besides, the traditional methods convert matrix equations into their equivalent forms by using the Kronecker product, which involve the inversion of the associated large matrix and result in increasing computation and excessive computer memory. The recently suggested gradient based iterative methods are more powerful; compare [10–13] and references therein.
The aim of the present paper is to derive representations of solutions to (1) and (2) and to estimate the norms of these solutions. Such estimates are important, in particular, for the investigations of linear and perturbations of nonlinear differential and difference equations. Norm estimates for solutions of the Sylvester equation whose coefficients are normal operators can be found in . In the finite dimensional case solution estimates for (4) and (6) have been established in .
In this paper, in particular, we considerably generalize the main results from .
Denote by and the spectrum and spectral radius of , respectively.
Theorem 1. Let be the roots taken with the multiplicities of the polynomialand letThen (1) has a unique solution which can be represented by and the series strongly converges.
The proof of this theorem is presented in the next section.
2. Proof of Theorem 1
Following [1, Section I.3], introduce the operators and by and , respectively. Besides, and commute.
Let be a scalar function regular on . Define the operator valued functionwhere , are closed Jordan contours surrounding and , respectively. So for any bounded linear operator Besides,for functions , regular on . Take . Ifthen due to formula from [1, Section I.3] (1) can be written asSincemaking use of (13), we can writeBut . Thus,and therefore we get the following result.
Furthermore, for , satisfyingconsider the operatorThe series converges andWe thus get the following result.
3. Representations of Solutions to (2)
Assume that is invertible; then from (2) we haveLet denote the lower spectral radius of : . Taking into account the fact that and applying now Theorem 1 to (26) with instead of , we get the following result.
Corollary 4. LetThen (2) has a unique solution which can be represented byand the series strongly converges.
Condition (27) does not allow us to consider the Lyapunov equation, since . Because of this we are going to derive the representation of solutions to (2) under other conditions. To this end putAccording to (19) for a solution to (26) we haveprovidedHere is defined as . ThusNow the invertibility condition for can be removed. We thus have proved the following result.
Lemma 6. Let the conditionhold. Then (4) has a unique solution , which can be represented as
Corollary 7. Let the conditionhold. Then the equationhas a unique solution , which can be represented as
4. Solution Estimates in the Finite Dimensional Case
4.1. Equation (1)
In this section is a Euclidean space. Let be the eigenvalues of a matrix counted with their multiplicities. The following quantity (the departure from normality of ) plays a key role hereafter:where is the Frobenius (Hilbert-Schmidt norm) of .
The following relations are checked in [15, Section 2.1]:where . If is a normal matrix: , then . If and are commuting matrices, then . By the inequality between geometric and arithmetic mean values we haveSo . Obviously .
If is normal, then andButThus, we haveprovided is normal. If both and are normal, thenFurthermore, obviously,Takingwe getwhereNow Corollary 9 implies the following result.
Theorem 1 and simple calculations imply the following result.
Corollary 11. Let . Then (7) has a unique solution , which satisfies the inequalities
4.2. Equation (2)
If is normal, then and thereforeIf both and are normal, then
For the Sylvester equation we have the following result.
Corollary 13. Let . Then (4) in has a unique solution , which satisfies the inequality
Corollary 14. Let . Then (5) in has a unique solution , which satisfies the inequality
5. Solution Estimates in the Infinite-Dimensional Case
5.1. Equation (1)
In this section , a separable Hilbert space. It is assumed thatPutRecall that and are the eigenvalues of a matrix counted with their multiplicities. If is a Hilbert-Schmidt operator, then , wherecompare Lemma 6.5.2 . Due to Example 7.10.2 from  we can writeAs it was shown in Section 4provided (9) holds. Hence, and from (80), it directly follows thatwhereNow Lemma 2 implies the following result.
5.2. Equation (2)
Due to Example 7.10.3 from  we haveprovided satisfies (77). Now let us consider (2), assuming that condition (42) holds. Due to (41) and (91),where . Thuswhereprovided . Due to Lemma 5 we arrive at the following result.
Let . Then, if is normal, then and thereforeIf both and are normal, thenAccording to Corollary 17 we get the following result.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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