Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. Moreover, we introduce the notion of weak kinematical similarity and prove a reducibility result by the spectral theorem.


Introduction
Let A k ∈ R N ×N , k ∈ Z, be a sequence of invertible matrices. In this paper, we consider the following nonautonomous linear difference equations , denote the evolution operator of (1.1), i.e., Φ(k, l) =
An invariant projector of (1.1) is defined to be a function P : Z → R N ×N of projections P k , k ∈ Z, such that for each P k the following property holds We say that (1.1) admits an exponential dichotomy if there exist an invariant projector P and constants 0 < α < 1, K ≥ 1 such that (1.2) Φ(k, l)P l ≤ Kα k−l , k ≥ l, The notion of exponential dichotomy was introduced by Perron in [28] and has attracted a lot of interest during the last few decades because it plays an important role in the study of hyperbolic dynamical behavior of differential equations and difference equations. For example, see [1,24,31] and the references therein. We also refer to the books [17,21,25] for details and further references related to exponential dichotomies. On the other hand, during the last decade, inspired both by the classical notion of exponential dichotomy and by the notion of nonuniformly hyperbolic trajectory introduced by Pesin (see [7]), Barreira and Valls have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way [8,9,10,11,12,13,14,15]. As explained by Barreira and Valls, in comparison to the notion of exponential dichotomies, nonuniform exponential dichotomy is a useful and weaker notion. A very general type of nonuniform exponential dichotomy has been considered in [5,6,19].
We say that (1.1) admits a nonuniform exponential dichotomy if there exist an invariant projector P and constants 0 < α < 1, K ≥ 1, ε ≥ 1, such that (1.4) Φ(k, l)P l ≤ Kα k−l ε l , k ≥ l, and (1.5) Φ(k, l)Q l ≤ K( 1 α ) k−l ε l , k ≤ l. When ε = 1, (1.4)-(1.5) become (1.2)-(1.3), and therefore a nonuniform exponential dichotomy becomes an exponential dichotomy. For example, given ω > a > 0, then the linear equation (1.6) u k+1 = e −ω+ak(−1) k −a(k−1)(−1) (k−1) u k , v k+1 = e ω−ak(−1) k +a(k−1)(−1) (k−1) v k admits a nonuniform exponential dichotomy, but does not admit an exponential dichotomy. In fact, we have Analogous arguments applied to the second equation yield the estimate (1.5). Moreover, when both k and l are even, we obtain the equality which means that the nonuniform part ε l = e 2al cannot be removed. Although the notion of nonuniform exponential dichotomy has been studied in a very wide range and many rich results have been obtained, up to now there are no results on the spectral theory of (1.1) in the setting of nonuniform exponential dichotomies. In this paper, we establish the spectral theory in the setting of strong nonuniform exponential dichotomies. We say that (1.1) admits a strong nonuniform exponential dichotomy if it admits a nonuniform exponential dichotomy with αε 2 < 1 in (1.4)-(1.5). For example, if ω > 5a, then (1.6) admits a strong nonuniform exponential dichotomy. We remark that the phrase "strong nonuniform exponential dichotomy" has been used in [8], however here we use this notion in a different sense. Moreover, [7,Theorem 1.4.2] indicates that the condition αε 2 < 1 is reasonable, which means that the constant ε belongs to the interval [1, 1/α).
Among the different topics on classical exponential dichotomies, the dichotomy spectrum is very important and many results have been obtained. We refer the reader to [2,3,18,26,29,30,32,34,35] and the references therein. The definition and investigation for finite-time hyperbolicity has also been studied in [16,22,23].
This paper is organized as follows. In Section 2 we propose a definition of spectrum based on strong nonuniform exponential dichotomies, which is called nonuniform dichotomy spectrum. Such a spectrum can be seen as a generalization of Sacker-Sell spectrum. We prove a nonuniform dichotomy spectral theorem. In Section 3 we prove a reducibility result for (1.1) using the spectral result. Recall that system (1.1) is reducible if it is kinematically similar to a block diagonal system with blocks of dimension less than N .

Nonuniform dichotomy spectrum
Consider the weighted system is its evolution operator. If for some γ ∈ R + , (2.1) admits a nonuniform exponential dichotomy with projector P k and constants K ≥ 1, 0 < α < 1 and ε ≥ 1, then P k is also invariant for (1.1), that is and the dichotomy estimates of (2.1) are equivalent to (2.2) Φ(k, l)P l ≤ K(γα) k−l ε l , k ≥ l,  Proof. For each γ ∈ ρ ED (A), the weighted system (2.1) admits an exponential dichotomy. Consequently, the weighted system (2.1) admits a strong nonuniform exponential dichotomy. Thus, γ ∈ ρ N ED (A), which implies that ρ ED (A) ⊂ ρ N ED (A), and therefore Σ N ED (A) ⊂ Σ ED (A). ✷ Let us define for γ ∈ ρ N ED (A) and where ε is the constant in (2.2)-(2.3). One may readily verify that S γ and U γ are invariant vector bundles of (1.1), here we say that a nonempty set W ⊂ Z × R N is an invariant vector bundle of (1.1) if (a) it is invariant, i.e., (l, ξ) ∈ W ⇒ (k, Φ(k, l)ξ) ∈ W for all k ∈ Z; and (b) for every l ∈ Z the fiber W(l) = {ξ ∈ R N : (l, ξ) ∈ W} is a linear subspace of R N . As a first glance, S γ and U γ are not well defined because they seem to depend on the constant ε, which may be not unique in (2.2)-(2.3). However, the following result ensures that S γ and U γ are well defined and they do not depend on the choice of the constant ε. First we recall that the invariant projector P is unique for (1.1) and (2.1) following the arguments in [21,Chapter 2]. Although the arguments in [21] are done in the setting of exponential dichotomies, it is not difficult to verify that they are also applicable to the case of nonuniform exponential dichotomies.
Lemma 2.2. Assume that (2.1) admits a strong nonuniform exponential dichotomy with invariant projector P for γ ∈ R + . Then Proof. We show only S γ = im P . The fact U γ = ker P is analog and the fact S γ ⊕ U γ = Z × R N is clear.
First we show S γ ⊂ im P. Let l ∈ Z and ξ ∈ S γ (l). Then there exists a positive constant C such that Φ(k, l)ξ ≤ Cγ k ε l , k ≥ l. We write ξ = ξ 1 + ξ 2 with ξ 1 ∈ imP l and ξ 2 ∈ kerP l . We show that ξ 2 = 0. The invariance of P implies for k ∈ Z, we have the identity Since (2.1) admits a strong nonuniform exponential dichotomy, the following inequality holds which implies that ξ 2 = 0 by letting k → ∞, since αε < 1.
is an invariant vector bundle which satisfies exactly one of the following two alternatives and the statements given in each alternative are equivalent: Alternative I Alternative II Now we are in a position to state and prove the nonuniform dichotomy spectral theorem which will be essential to prove the reducibility result in Section 3. The proof follows the idea and technique of the classical dichotomy spectrum proposed in [33], we present the details for the reader's convenience.
Then the sets then for every i = 1, . . . , n − 1 the intersection The invariant vector bundles W i , i = 0, . . . , n + 1, are called spectral bundles and they are independent of the choice of γ 0 , . . . , γ n in (2.4), (2.5) and (2.6). Moreover Proof. Recall that the resolvent set ρ N ED (A) is open and therefore Σ N ED (A) is the disjoint union of closed intervals. Next we will show that Σ N ED (A) consists of at most N intervals. Indeed, if Σ N ED (A) contains N + 1 components, then one can choose a collections of points ζ 1 , . . . , ζ N in ρ N ED (A) such that ζ 1 < · · · < ζ N and each of the intervals (0, Assume that x k+1 = 1 ζN A k x k admits a strong nonuniform exponential dichotomy with invariant projector P ≡ Id, then x k+1 = 1 ζ A k x k also admits a strong nonuniform exponential dichotomy with the same projector for every ζ > ζ N . Now we have the conclusion (ζ N , ∞) ⊂ ρ N ED (A), which is a contradiction. This proves the alternatives for Σ N ED (A).
On the other hand, using a similar argument as in equations (1.6), we know that the nonuniform part ε l cannot be removed in the estimates (2.11) and (2.13). Therefore, (2.8) does not admit an exponential dichotomy, which means that Σ ED (A) = R + . ✷

Reducibility
In this section we employ Theorem 2.6 to prove a reducibility result. For the reducibility results in the setting of an exponential dichotomy, we refer the reader to [20,27,35] and the references therein.
Proof. Let n ∈ Z be arbitrary but fixed. Note that the rank of the projector P n is independent of n ∈ Z (see [16, Page 1100]), then there exists a nondegenerate matrix T ∈ R N ×N such that with N 1 = dim imP and N 2 = dim kerP . Define On the other hand, we have Φ(k, l)P l = Φ(k, n)Φ(n, l)P l = Φ(k, n)P n Φ(n, l) = Φ(k, n)P n Φ −1 (l, n) .
The change of variables x k = S k y k then transforms (1.1) into (3.5).
The next lemma is important to establish the reducibility results and its proof follows along the lines of the proof of Siegmund [35]. See also Coppel [21] and Aulbach et al. [2] Lemma 3.3. [21, Chapter 5] Let P be an orthogonal projection (P T = P ) and let X be an invertible matrix. Then there exists an invertible matrix function S : Z → R N ×N such that Then the mapping is a positive definite, symmetric matrix for every k ∈ Z. Moreover there is a unique function R : Z → R N ×N of positive definite symmetric matrices R k , k ∈ Z, with R 2 k = R k , P R k = R k P . We remark that S −1 k in Lemma 3.3 is bounded in the setting of an exponential dichotomy. However, in the setting of a nonuniform exponential dichotomy, S −1 k can be unbounded, because Φ(k, k)P k ≤ Kε k for k ≥ 0. To overcome the difficulty, we introduce a new version of non-degeneracy, so-called weak non-degeneracy and define the concept of weak kinematical similarity. Some results will be obtained on the decoupling into two blocks which will play an important role in the analysis of reducibility.  For the sake of comparison, we denote kinematical similarity by (1.1) ∼ (3.5) or A k ∼ B k . Definition 3.6. We say that equation (1.1) is reducible, if it is weakly kinematically similar to equation (3.5) whose coefficient matrix B k has the block form where B 1 k and B 2 k are matrices of smaller size than B k . The following theorem shows that if (1.1) admits a nonuniform exponential dichotomy, then there exists a weakly non-degenerate transformation such that A k w ∼ B k and B k has the block form (3.6), i.e., system (1.1) is reducible.
Theorem 3.7. Assume that (1.1) admits a nonuniform exponential dichotomy (not necessary strong) of the form (3.1)-(3.2) with invariant projector P k = 0, Id. Then (1.1) is weakly kinematically similar to a decoupled system for some locally integrable matrix functions where N 1 := dim imP and N 2 := dim kerP . That is, system (1.1) is reducible.
Thus, S is weakly non-degenerate. Setting where R k is defined in Lemma 3.3 and X k = S k R k . Obviously, R k is the fundamental matrix of linear system Now we need to show that A k w ∼ B k and B k has the block diagonal form First, we show that A k w ∼ B k . In fact, Now we show that system (1.1) is weakly kinematically similar to (3.7). By Lemma 3.3, R k+1 and R −1 k commute with the matrixP for every k ∈ Z. It follows that Identity (3.8) implies that Therefore B 3 k ≡ 0 and B 4 k ≡ 0. Thus B k has the block form Now the proof is finished. ✷ From Theorem 3.7, we know that if (1.1) admits a nonuniform exponential dichotomy, then there exists a weakly non-degenerate transformation S k such that A k w ∼ B k and B k has two blocks of the form (3.6).
Lemma 3.8. Assume that (1.1) admits a nonuniform exponential dichotomy with the form of estimates (3.1)-(3.2) and rank(P ) = N 1 , (0 < N 1 < N ), and there exists a weakly non-degenerate transformation S k such that A k w ∼ B k . Then system (3.5) also admits a nonuniform exponential dichotomy, and the projector has the same rank.
Proof. Suppose that S k is weakly non-degenerate, which means that there exists M = M (ε) > 0 such that S k ≤ M ε |k| and S −1 k ≤ M ε |k| and such that A k w ∼ B k . Let X k = S k Y k . It is easy to see that Y k is the fundamental matrix of system (3.5). To prove that system (3.5) admits a nonuniform exponential dichotomy, we first consider the case k ≥ l and obtain where M 1 = M ε 2|l| . Similar argument shows that (3.9) and (3.10), it is easy to see that system (3.5) admits a nonuniform exponential dichotomy. Clearly, the rank of the projector is k. ✷ Lemma 3.9. Assume that the systems (1.1) and (3.5) are weakly kinematically similar via S k . If for a constant γ ∈ R + the system (2.1) admits a strong nonuniform exponential dichotomy with constants K > 0, 0 < α < 1, ε ≥ 1 and invariant projector P , then the system (3.11) y k+1 = 1 γ B k y k also admits a strong nonuniform exponential dichotomy.
Proof. Obviously, P is also an invariant projector for (1.1). The dichotomy estimates are equivalent to Therefore, (3.11) admits a strong nonuniform exponential dichotomy. ✷ The following result follows directly from Lemma 3.9. Theorem 3.11 (Reducibility Theorem). Assume that (1.1) admits a strong nonuniform exponential dichotomy. Due to Theorem 2.6, the dichotomy spectrum is either empty or the disjoint union of n closed spectral intervals I 1 , . . . , I n with 1 ≤ n ≤ N , i.e., Σ N ED (A) = ∅ (n = 0) or Σ N ED (A) = I 1 ∪ · · · ∪ I n .
Then there exists a weakly kinematic similarity action S : Z → R N ×N between (1.1) and a block diagonal system Proof. If for any γ ∈ R + , system (2.1) admits a strong nonuniform exponential dichotomy, then Σ N ED (A) = ∅. Conversely, for any γ ∈ R + , system (2.1) does not admit a strong nonuniform exponential dichotomy, then Σ N ED (A) = R + . Now, we prove the theorem for the nontrivial case (Σ N ED (A) = ∅ and Σ N ED (A) = R + ).
Recall that the resolvent set ρ N ED (A) is open and therefore the dichotomy spectrum Σ N ED (A) is the disjoint union of closed intervals. Using Theorem 2.6, we can assume If I 1 = [a 1 , b 1 ] is a spectral interval, then (0, γ 0 ) ⊂ ρ N ED (A) and W 0 = S γ0 for some γ 0 < a 1 due to Theorem 2.6, which implies that γ0 A k x k admits a strong nonuniform exponential dichotomy with an invariant projector P 0 . By Theorem 3.7 and Corollary 3.10, there exists a weakly non-degenerate transformation N 0 due to Theorem 3.7, Lemma 2.2 and Theorem 2.6. If I 1 = (0, b 1 ] is a spectral interval, a block B 0 k is omitted. Now we consider the following system By using Lemma 2.5, we take γ 1 ∈ (b 1 , a 2 ). In view of (b 1 , a 2 ) ⊂ ρ N ED (B 0, * k ), γ 1 ∈ ρ N ED (B 0, * k ), which implies that Then A k w ∼ A 1 k and A 1 k has three blocks of the form Applying similar procedures to γ 2 ∈ (b 2 , a 3 ), γ 3 ∈ (b 3 , a 4 ), . . ., we can construct a weakly non-degenerate transformation x k = S k x Finally, we show that N i = dim W i . From the claim above, we note that dim B 0 k = dim W 0 , dim B 1 k ≥ dim W 1 , . . . , dim B n k ≥ dim W n , dim B n+1 k = dim W n+1 and with Theorem 2.6 this gives dim W 0 +· · ·+dim W n+1 = N , so dim B i k = dim W i for i = 0, . . . , n + 1. Now the proof is finished. ✷