Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

Abstract: The paper is devoted to the discrete Lyapunov equation X− A∗XA = C, where A and C are given operators in a Hilbert spaceH and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations inH. Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.


Introduction and Notations
Let H be a complex separable Hilbert space with a scalar product (., .), the norm .= (., .), and unit operator I = I H .By B(H), we denote the set of all bounded linear operators in H.In addition, Ω denotes the unit circle: Ω = {z ∈ C : |z| = 1}.An operator A is said to be Schur-Kohn stable, or simply stable, if its spectrum σ(A) lies inside Ω.Otherwise, A will be called an unstable operator.
Consider the discrete Lyapunov equation: where A, C ∈ B(H) are given operators and X should be found.That equation arises in various applications, cf.[1].Sharp norm estimates for solutions of (1) with Schur-Kohn stable finite dimensional and some classes of infinite dimensional operators have been derived in [2,3].At the same time, to the best of our knowledge, norm estimates for solutions of (1) with unstable A have not been obtained in the available literature.
Our aim in the present paper is to establish sharp norm estimates for solutions of Equation ( 1) with an unstable operator A. In addition, we refine and complement estimates for (1) with stable operator coefficients from [2,3].
The point estimates enable us to suggest new dichotomy conditions for nonautonomous linear difference equations and explicit stability conditions for the nonautonomous nonlinear difference equations in a Hilbert space.
The dichotomy of various abstract difference equations has been investigated by many mathematicians, cf.[4] and [5][6][7][8][9][10][11] and the references therein.In particular, the main result of the paper [8] gives a decomposition of the dichotomy spectrum considering the upper dichotomy spectrum, lower dichotomy spectrum, and essential dichotomy spectrum.In addition, in [8], it is proven that the dichotomy spectrum is a disjoint union of closed intervals.In [9,11], an approach concerning the characterization of the exponential dichotomy of difference equations by means of an admissible pair of sequence Banach spaces has been developed.The paper [12] considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces.These concepts use two types of dichotomy projection sequences and generalize some well-known dichotomy concepts.
Certainly, we could not survey here all the papers in which in the general situation the dichotomy conditions are formulated in terms of the original norm.We formulate the dichotomy conditions in terms of solutions of Lyapunov's equation.In appropriate situations, that fact enables us to derive upper and lower solution estimates.In addition, traditionally, the existence of dichotomy projections is assumed.We obtain the existence of these projections via perturbations of operators.
The stability theory for abstract nonautonomous difference equations has a long history, but mainly linear equations have been investigated, cf.[13][14][15] and the references therein.Regarding the stability of nonlinear autonomous difference equations in a Banach space, see [16].The stability theory for nonlinear nonautonomous difference equations in a Banach space is developed considerably less than the one for linear and autonomous nonlinear equations.Here, we should point out the paper [17], in which the author studied the local exponential stability of difference equations in a Banach space with slowly-varying coefficients and nonlinear perturbations.Besides, he established the robustness of the exponential stability.Regarding other results of the stability of nonlinear nonautonomous difference equations in an infinite dimensional space, see for instance [2], Chapter 12.
In this paper, we investigate semilinear nonautonomous difference equations in a Hilbert space and do not require that the coefficients are slowly varying.
Introduce the notations.For an The Schatten-von Neumann ideal of compact operators A in H with the finite Schatten-von Neumann norm N p (A) := (trace (A * A) p/2 ) 1/p (1 ≤ p < ∞) is denoted by SN p .In particular, SN 2 is the Hilbert-Schmidt ideal and N 2 (.) is the Hilbert-Schmidt norm.

Auxiliary Results
In the present section, we have collected norm estimates for powers and resolvents of some classes of operators and estimates for the powers of their inverses.They give us bounds for the solution of Equation (1).

Operators in Finite Dimensional Spaces
Let H = C n (n < ∞) be the complex n-dimensional Euclidean space and C n×n be the set of complex n × n matrices.In this subsection, A ∈ C n×n ; λ k (A), k = 1, ..., n, are the eigenvalues of A, counted with their multiplicities.Introduce the quantity (the departure from normality of A): The following relations are checked in [3], Section 3.1: If A is a normal matrix: AA * = A * A, then g(A) = 0.
Due to Example 3.3 from [3]: ( Recall that 1 It is attained for a normal operator A, since g(A) = 0, 0 0 = 1, and A m = r m s (A) in this case.By Theorem 3.2 from [3]: This inequality is also attained for a normal operator.Now, let r l > 0.Then, by Corollary 3.6 from [3], Inequality ( 4) is equality if A is a normal operator.In addition, by Theorem 3.3 of [3] for any invertible A ∈ C n×n and 1 ≤ p < ∞, one has: and: Hence, and: Now, (2) and (5) imply:

Hilbert-Schmidt Operators
In the sequel, H is infinite dimensional.In this subsection, A is in SN 2 and: where λ k (A) (k = 1, 2, ...) are the eigenvalues of A ∈ B(H), counted with their multiplicities and enumerated in the nonincreasing order of their absolute values.Since: If A is a normal Hilbert-Schmidt operator, then g(A) = 0, since: in this case.Moreover, cf. [3], Section 7.1.Due to Corollary 7.4 from [3], for any A ∈ SN 2 , we have: This inequality and Inequality (9) below are attained for a normal operator.Furthermore, by Theorem 7.1 from [3], for any A ∈ SN 2 , we have: By the Schwarz inequality: Taking c 2 = 1/2, from (9), we arrive at the inequality:

Schatten-von Neumann Operators
In this subsection, A ∈ SN 2p for an integer p ≥ 1. Making use of Theorems 7.2 and 7.3 from [3], we have: (11) and: Since, the condition A ∈ SN 2p implies A − A * ∈ SN 2p , and one can use estimates for the resolvent presented in the next two subsections.

Noncompact Operators with Hilbert-Schmidt Hermitian Components
In this subsection, we suppose that: To this end, introduce the quantity: Furthermore, by Theorem 9.1 from [3], under Condition ( 14), we have, and: Now, let r l > 0.Then, by ( 16): Similarly, by (17): Let us point out an additional estimate for A −m .
Note that A −1 can be estimated by ( 18) and ( 19).

Noncompact Operators with Schatten-von Neumann Hermitian Components
In this subsection, it is assumed that: By Theorem 9.5 of [3], for any quasinilpotent operator V ∈ SN p , there is a constant b p dependent on p only, such that N p (V + V * ) ≤ b p N p (V − V * ).According to Lemma 9.5 from [3], b p ≤ p 2 e 1/3 .Put: Therefore, From the Weyl inequalities ([3], Lemma 8.7), we have N 2p (D I ) ≤ N 2p (A I ).Thus: If A has a real spectrum, then: We need the following result ([3], Theorem 9.5).

The Discrete Lyapunov Equation with a Stable Operator Coefficient
Theorem 2. Let A ∈ B(H) and r s (A) < 1.Then, for any C ∈ B(H), there exists a linear operator X = X(A, C), such that: Moreover, and: Thus, if C is strongly positive definite, then X(A, C) is strongly positive definite.
For the proof of this theorem and the next lemma, for instance see [1] ([2], Section 7.1).
Lemma 2. If Equation (32) with C = C * > 0 has a solution X(A, C) > 0, then the spectrum of A is located inside the unit disk.
Due to Representations (33) and (34), we have: and: respectively.From the latter inequality, it follows Similar results can be found in the Exercises of Chapter 1 from [18].Again, assume that Condition (27) holds.Then, ).Now, (36) implies: If A is normal, then A k = r k s (A), and (35) yields: Example 1.Let A ∈ C n×n .Then, (2) and (35) yield: Note that if A is normal, then g(A) = 0, and Example 3.3 gives us Inequality (38).Let us point to the more compact, but less sharper estimate for X(A, C).Making use of ( 3) and (37), we can assert that: Then, ( 8) and (35) yield: If A is normal, then this example gives us Inequality (38).Furthermore, (37) and (10) imply: Example 3. Assume that A I ∈ SN 2 .Then, (4) and (35) yield: If A is normal, hence we get (38).Inequality (37) along with ( 16) and (17) give us the inequalities: respectively.For a self-adjoint operator S, we write S ≥ 0 (S > 0) if it is positive definite (strongly positive definite).The inequalities S ≤ 0 and S < 0 have a similar sense.
Note that (33) gives a lower bound for X(A, C) with C = C * ≥ 0. Indeed, If C is noninvertible, then r l (C) = 0, and: if the corresponding operator is invertible.Therefore, we arrive at Lemma 3. Let X(A, I) = X(A) be a solution of (32) with C = I and r s (A) < 1. Then: Therefore, X −1 (A) ≤ 1 in the general case.

Operators with Dichotomic Spectra
In this section, it is assumed that σ(A) is dichotomic.Namely, where σ ins and σ out are nonempty nonintersecting sets lying inside and outside Ω, respectively: sup |σ ins | < 1 and inf |σ out | > 1. Put: Therefore, P is the Riesz projection of A, such that σ(AP) = σ ins and σ(A(I − P)) = σ out .We have A = A ins + A out , where A ins = AP = PA, A out = (I − P)A = A(I − P).
The analogous results can be found in ( [18], Exercises of Chapter 1).

Linear Autonomous Difference Equation
In this section, we illustrate the importance of solution estimates for (32) in the simple case.To this end, consider the equation: is dichotomic, if there exist a projection P = 0, P = I and constants ν ∈ (0, 1), µ > 1 and a, b > 0 such that u k ≤ aν k u 0 if u 0 ∈ PH and u k ≥ mµ k u 0 if u 0 ∈ (I − P)H.

Perturbations of Operators
To investigate nonautonomous equations, in this section, we consider some perturbations of operators.

The Case r l
and X = X(A) be the solution of (62).If, in addition, then with Y = −X(A), one has: Proof.With Z = Ã − A, one has: Since Y is positive definite, hence, by ( 68), as claimed.

Perturbation of Operators with Dichotomic Spectra
Let Condition (50) hold, and: is fulfilled and: Therefore, Ω ∩ σ( Ã) = ∅.Moreover, Ã has a dichotomic spectrum: where σins and σout are nonempty nonintersecting sets lying inside and outside Ω, respectively.Indeed, let Hence, (70) follows from (50) and the semi-continuity of the spectrum.Put: Ãins = P Ã and Ãout = (I − P) Ã.With the notations of Section 5, According to (69) with q = A − Ã , we obtain: , one has: In this section, X ins and X out are solutions of the equations of (55), (56), respectively, with C = I; i.e., and: X out − A * out X out PA out = (I − P * )(I − P).
Making use of Lemma 8.2, we get: then with Y out = −X out , one has: where m out = 1 − 2 X out q out A out .

Stability
Consider the equation: with given u 0 ∈ H.For some A ∈ B(H), define the norms: where X = X(A) is the solution of (62).Throughout this section and the next one, it is assumed that sup k A k < ∞ and denoted q 0 := sup k A − A k .Theorem 4. Let there be an A ∈ B(H) with r s (A) < 1, such that: Then, for any solution of u k of (72), one has: where a 0 := 1 − (2q 0 X + q 2 0 ).
Hence, each A k has a dichotomic spectrum: where σ ins (A k ) and σ out (A k ) are nonempty nonintersecting sets lying inside and outside Ω, respectively.Put: A k,ins = P k A k and A k,out = (I − P k )A k .With A ins defined as Section 5, According to (82): ), one has: In this section, X ins and X out are solutions of Equation ( 71) and the equation X out − A * out X out PA out = (I − P * )(I − P), respectively.If: then Corollary 8.3 implies: where: c 0,ins := 1 − X ins (2q 0,ins A ins + q 2 0,ins ).
Similar results for the periodic equations in the finite-dimensional space were established in the article [19].
Lemma 12. Let Condition (95) hold with = ∞.Let there be an A ∈ B(H) with r s (A) < 1 and: where X is the solution of (62).Then: for any solution u k of (94).
Then, the zero solution to (94) is exponentially stable.