Journal of Mathematics
Volume 2015 (2015), Article ID 524829, 7 pages
http://dx.doi.org/10.1155/2015/524829

Research Article

Norm Estimates for Solutions of Polynomial Operator Equations

Michael Gil’

Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

Received 20 July 2015; Accepted 31 August 2015

Academic Editor: William E. Fitzgibbon

Copyright © 2015 Michael Gil’. 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 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 equations

where ,   , , , 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 [1] and [2]. This tradition was continued in [3], which deals with the equation

Clearly, (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 equation

is the famous Lyapunov equation playing an essential role in the stability theory of differential equations. The equation

is an example of (1); in particular, the discrete Lyapunov equation

is an important tool in the theory of difference equations.

About other operator equations see [7–9] and references therein. In particular, [9] 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 [8] 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 [2]. In the finite dimensional case solution estimates for (4) and (6) have been established in [14].

In this paper, in particular, we considerably generalize the main results from [14].

Denote by and the spectrum and spectral radius of , respectively.

Theorem 1. Let    be the roots taken with the multiplicities of the polynomial

and let

Then (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 function

where , are closed Jordan contours surrounding and , respectively. So for any bounded linear operator

Besides,

for functions , regular on . Take . If

then due to formula from [1, Section I.3] (1) can be written as

Since

making use of (13), we can write

But . Thus,

and therefore we get the following result.

Lemma 2. Under condition (14) equation (1) has a unique solution defined by

Furthermore, for , satisfying

consider the operator

The series converges and

We thus get the following result.

Lemma 3. Let condition (20) hold. Then operator defined by (21) is the unique solution to the equation and .

Proof of Theorem 1. Put

Lemma 2 implies . Due to the previous lemma are solutions to the equations

provided (9) holds. So

Continuing this process for , according to Lemma 2, we prove the theorem.

3. Representations of Solutions to (2)

Assume that is invertible; then from (2) we have

Let 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. Let

Then (2) has a unique solution which can be represented by

and 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 put

According to (19) for a solution to (26) we have

provided

Here is defined as . Thus

Now the invertibility condition for can be removed. We thus have proved the following result.

Lemma 5. Let the condition

hold. Then (2) has a unique solution which can be represented by (32).

Lemma 6. Let the condition

hold. Then (4) has a unique solution , which can be represented as

Proof. For the brevity put and . Then and Moreover, . Equation (4) is equivalent to the following one:

Due to Theorem 9.2 from [2] a solution of (36) is defined by the equality

as claimed.

Corollary 7. Let the condition

hold. Then the equation

has a unique solution , which can be represented as

A solution of (39) is also given by . So under condition (38) we have

Now assume that

Then by (41)

Let . Then . So condition (42) means that .

Put

Then . Due to the previous lemma are solutions to the equations

provided (42) holds. So

Continuing this process for , according to Lemma 5, we obtain

We arrive at the following result.

Theorem 8. Let be the roots of and let condition (42) hold. Then (2) has a unique solution , which can be represented by (47).

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 have

So . Obviously .

By Corollary 2.7.2 from [15], one has

Note that . Furthermore, due to Example 1.10.3 from [15] we have

From (21) and Lemma 3 we have

provided condition (20) holds. Hence, and from (51), it directly follows

where

Now Lemma 2 and (54) imply the following result.

Corollary 9. Let condition (9) hold. Then a unique solution to (1) in satisfies the inequality

If is normal, then and

But

Thus, we have

provided is normal. If both and are normal, then

Furthermore, obviously,

Taking

we get

where

Now Corollary 9 implies the following result.

Corollary 10. Let condition (9) hold. Then a unique solution of (1) in satisfies the inequality

Theorem 1 and simple calculations imply the following result.

Corollary 11. Let . Then (7) has a unique solution , which satisfies the inequalities

From Corollary 11 and (63) it follows that

4.2. Equation (2)

Due to (41) and (52) under (38) we have

where . But

Thus

where

provided . Due to Lemma 5 we arrive at the following result.

Corollary 12. Let condition (42) hold. Then a unique solution to (2) in satisfies the inequality

If is normal, then and therefore

If 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

Consider the Lyapunov equation (5). Taking in Corollary 13   instead of and instead of , since and , we get the following result.

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 that

Put

Recall that and are the eigenvalues of a matrix counted with their multiplicities. If is a Hilbert-Schmidt operator, then , where

compare Lemma 6.5.2 [15]. Due to Example 7.10.2 from [15] we can write

As it was shown in Section 4

provided (9) holds. Hence, and from (80), it directly follows that

where

Now Lemma 2 implies the following result.

Corollary 15. Let conditions (9) and (77) hold. Then the unique solution to (1) in satisfies the inequality

If is normal, then and according to (59)

If both and are normal, then

Furthermore, according to (63), we get , where

where is defined by (62). Now Corollary 15 implies the following result.

Corollary 16. Let conditions (9) and (77) hold. Then a unique solution of (1) in satisfies the inequality

By virtue of Corollary 15 we can assert that a unique solution of (7) satisfies the inequality

provided satisfies conditions (77) and . From Corollary 16 it follows that

5.2. Equation (2)

Due to Example 7.10.3 from [15] we have

provided satisfies (77). Now let us consider (2), assuming that condition (42) holds. Due to (41) and (91),

where . Thus

where

provided . Due to Lemma 5 we arrive at the following result.

Corollary 17. Let conditions (42) and (77) hold. Then the unique solution to (2) in satisfies the inequality

Let . Then, if is normal, then and therefore

If both and are normal, then

According to Corollary 17 we get the following result.

Corollary 18. Let satisfies conditions (77) and . Then the solution to the Lyapunov equation (5) in satisfies the inequality

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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