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DICHOTOMY SPECTRA OF TRIANGULAR EQUATIONS

. Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant linear equations having triangular coeﬃcient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. This is surpris-ing because such a behavior diﬀers from both the half line situation, as well as the classical autonomous and periodic cases. At the same time triangular problems occur in various applications and particularly numerical techniques. Based on operator-theoretical tools, this paper provides various suﬃcient and veriﬁable criteria to obtain a corresponding diagonal signiﬁcance for ﬁnite- dimensional diﬀerence equations in the following sense: Spectral and continuity properties of the diagonal elements extend to the whole triangular system.


1.
Introduction. At least in finite dimensions, the local behavior of dynamical systems near constant or periodic solutions is generically determined by the spectrum of its linearization, i.e. by eigenvalues or Floquet multipliers. Provided the (Floquet) spectrum is disjoint from the stability boundary (the unit circle in discrete time resp. the imaginary axis in continuous time) one speaks of hyperbolicity. When extending this setting and dealing with general nonautonomous systems or aperiodic solutions, hyperbolicity is not a generic property anymore and cannot be characterized in terms of eigenvalues. On the one hand, this led to the theory of Lyapunov exponents, where the Multiplicative Ergodic Theorem (MET for short, cf., e.g. [27]) yields an ambient "nonautonomous linear algebra." On the other hand, especially for systems being irregular in the sense of Lyapunov (see [8]), a further appropriate spectral notion is given in terms of the coarser dichotomy (dynamical, or Sacker-Sell) spectrum Σ ⊆ R (cf. [41,27,6]). This concept is particularly suitable to obtain stability information, and far beyond that to develop a geometric theory for time-dependent equations involving invariant manifolds and topological linearizations [28], as well as normal forms [43,44]. In addition, it turned out to be beneficial to investigate several dynamically relevant subsets of the dichotomy spectrum for the following reasons: (1) These sets allow to classify nonautonomous bifurcations on a linear level [37]. (2) While Σ is only upper-semicontinuous under general bounded perturbations, appropriate relations between its dichotomy subspectra yield even continuity for Σ (see [39]).
Typically, the dichotomy spectrum is only accessible on a numerical basis. As a result, both the approximate computation (cf. [13,24]), and also further properties (see [37]) of Σ received attention over the recent years. Indeed, many of the computational methods are based on the strategy to transform a linear difference or differential equation to triangular form without affecting its spectrum (or stability properties), and then to extract the spectrum from their resulting diagonal: Since the diagonal elements are scalar functions, their dichotomy spectra are single intervals whose boundary consists of lower and upper Bohl exponents. The spectrum of the whole system then results as the union of the diagonal spectra. This is a valid technique, as long as the equations are merely dichotomic on a half line. Nonetheless, when dealing with exponential dichotomies and related spectra on the whole time axis, also information on elements off the diagonal is needed, or specific assumptions on the diagonal are necessary. Besides numerical techniques, another source for (block-) triangular linear problems are variational equations related to extinction equilibria of nonlinear models in e.g. population dynamics (we refer to [25]). From a theoretical perspective, also the proof of the MET in [27] rests upon a triangulation technique.
Indeed, the full axis dichotomy spectrum has more subtle (and weaker) perturbation properties than the related half line concept. This is due to the fact that a dichotomy on the entire axis requires dichotomies on both semiaxes and beyond that a stable subspace X + (in forward time) being complementary to the unstable subspace X − (in backward time). Thus, for instance the approximation algorithm in [14] has to verify the latter geometrical condition. Roughly speaking, this is achieved "Evans function-like" by choosing bases of X + and X − , and showing the linear independence of their union. An alternative approach to obtain the subspaces X − , X + in discrete time is given in [23].
These observations motivate our deeper analysis of spectral properties for w.l.o.g. (block) upper-triangular linear nonautonomous dynamical systems. For this purpose, the paper restricts to time-dependent finite-dimensional difference equations, since they provide a setting tailor-made to apply convenient operator-theoretical tools as previously exemplified in [10,5] or [36,37,39]. This approach allows rather short proofs and can be seen as the main conceptional contribution of the present paper. Yet, we do not conceal that it somehow underplays the importance of X − , X + being complementary. Our presentation begins in the subsequent Sect. 2 with preparations on characteristic, Lyapunov and Bohl exponents, as well as exponential dichotomies in discrete time. These concepts are illustrated by several examples to which we will return throughout the text. We also emphasize the close and central relationship between the dichotomy spectrum and weighted shift operators on the Hilbert space of square-summable sequences. Actually, this observation is crucial in order to apply corresponding results from operator theory and therefore for the entire paper. The following Sects. 3-4 illuminate how results on the spectra of abstract triangular operators provide sufficient conditions on the diagonal sequences, as well as on the off-diagonal elements, such that our desired diagonal significance holds: This means • the dichotomy spectrum and its dynamically relevant subspectra are determined by the union of the corresponding diagonal spectra (cf. Sect. 4.2), • continuity of the diagonal spectra w.r.t. the Hausdorff distance yields continuity of the full spectrum (cf. Sect. 4.4); a precise definition follows below. Among others, these conditions are based on ambient compatibility conditions comparing a system's growth in forward and backward time by means of their Lyapunov filtrations (cf. [8]). For instance, Sect. 3 tackles the basic situation of diagonal systems, whereas Sect. 4 studies upper blocktriangular equations. Sufficient conditions for diagonal significance depending on the diagonal systems, or the off-diagonal entries are provided. The obtained prototype results extend to triangular equations by means of inductive arguments, which can be found in Sect. 5. An explicit example from population dynamics is presented in Sect. 6 in order to illustrate our results. Finally, for the reader's convenience the paper closes with appendices covering the required basics of operator theory and matrix-weighted shifts.
Although the present paper sticks to a discrete time situation, we are about to explain in Sect. 7 to what extend the results are useful in an ODE context as well. In addition, we recently became aware of Flaviano Battelli's and Ken Palmer's preprint [9] dealing with dichotomies and the related spectrum of block-triangular equations in continuous time. They allow unbounded coefficients and obtain also necessary conditions for diagonal significance in the dichotomy spectrum. Moreover, a procedure to determine the full-axis spectrum from the half line spectra is given. The methods in [9] are different from ours though.
We start with the necessary terminology: Given a real interval I ⊆ R, one denotes an intersection I Z := I ∩Z with the integers Z as discrete interval ; for such a discrete interval I, set I := {k ∈ Z : k + 1 ∈ I}. Here, I will typically be unbounded, and e.g. of the form Z + κ := [κ, ∞) Z , Z − κ := (−∞, κ] Z κ ∈ Z, or Z. Let us write K for one of the fields R or C. On K d we denote the Euclidean resp. unitary norm by |·|, write L(K d ) for the d × d-matrices and GL(K d ) for the invertible matrices. The space of square-summable sequences in K d is abbreviated as 2 = 2 (K d ) throughout.
Let K(K) denote the family of nonempty compact subsets of K and be the Hausdorff distance. Then the pair (K(K), h) becomes a metric space. Finally, the closure of a subset M ⊆ K d is denoted by M , and M • is its interior.

2.
Preliminaries. Consider a linear nonautonomous difference equation We often identify (∆ A ) with the matrix sequence A = (A k ) k∈I in the Banach space of bounded matrix sequences. The solutions to (∆ A ) can be expressed in terms of the transition matrix Along with (∆ A ) let us introduce the adjoint difference equation The following subsections collect some basic notions, establish our terminology and provide elementary examples necessary for the remaining paper: 2.1. Characteristic exponents and Lyapunov filtration. Assume that the discrete interval I is unbounded above. In order to capture the long-term behavior of (∆ A ) consider the (upper) characteristic exponent of its solution starting in x ∈ K d ; this exponent is independent of the initial time κ ∈ I and clearly fulfills χ A (0) = 0. A difference eqn. (∆ A ) possesses up to d characteristic exponents which form its (upper) Lyapunov spectrum with n ≤ d. We suppose that the positive reals λ j are ordered according to 0 < λ 1 < . . . < λ n .
The sublevel sets W j := x ∈ K d : χ A (x) ≤ λ j are linear subspaces of K d yielding the Lyapunov filtration of strict inclusions Concerning this, and more details on Lyapunov spectra we refer to [8, pp. 56ff].
For the adjoint difference eqn. (∆ * A ) the characteristic exponent is defined by As above one obtains a finite Lyapunov spectrum Besides characteristic exponents and Lyapunov filtrations, a further and arguably more appropriate tool to capture the asymptotics of nonautonomous equations are exponential dichotomies.
Given an unbounded discrete interval I, a linear difference eqn. (∆ A ) has an exponential dichotomy on I (ED for short, cf. [19,6]), if there exists a sequence of projections P k ∈ L(K d ), k ∈ I, with P k+1 A k = A k P k for all k ∈ I , growth rates α ∈ (0, 1) and a constant K ≥ 1 such that the estimates and k, l ∈ I hold. Then dichotomy spectrum of (∆ A ) is defined as The invertibility assumption on A k ensures that an empty spectrum or a spectral interval (0, β m ] can be avoided precisely in case Due to its role for stability properties, max Σ(A) is called stability radius of (∆ A ). We speak of a discrete spectrum, if Σ(A) is finite. Discrete spectra on the half line I = Z + κ typically occur for asymptotically periodic equations, whereas on the whole line I = Z, periodic (or autonomous) equations possess a discrete spectrum.
If we denote the dichotomy spectra associated with the discrete intervals Z + κ , Z − κ or Z by Σ + (A), Σ − (A) resp. Σ(A), then the inclusions hold (see [28,p. 88,Thm. 5.13]). Provided (∆ A ) can be described using a discrete-time skew-product flow over a compact base, it was established in [27] that the Lyapunov spectrum is contained between Σ and its boundary ∂Σ. Generally speaking the Lyapunov spectrum is finer than the dichotomy spectra, and we refer to our concluding Ex. 2.6 for concrete examples illustrating these inclusions. For the family of discrete intervals J ⊆ I with fixed length n ∈ N one writes I n := {J ⊆ I : J is a discrete interval with #J = n} for all n ∈ N. It has advantages to introduce Lyapunov and Bohl exponents abstractly: Suppose thereto that A is a normed unital algebra over K ∈ {R, C} with norm |·|. Let us define the lower resp. upper Bohl exponent of a sequence a = (a k ) k∈I in A as and independent of κ ∈ I, as long as all a j ∈ A, j ∈ I, are invertible. It goes without saying that I has to be unbounded above in order to introduce λ + (a), λ + (a), while the definition of λ − (a), λ − (a) only makes sense for I being unbounded below. When dealing with Bohl exponents β I (a), β I (a) it suffices that I is solely unbounded.
Remark 2.1. In the Banach algebra A = K the Lyapunov exponents of a sequence a and the characteristic exponents of the corresponding scalar difference equation are related by Properties of Lyapunov and particularly Bohl exponents, as well as the fact that the limits in (2.2) exist under natural assumptions, are given in Proposition 2.2. On unbounded discrete subintervals J ⊆ I one has and the positive homogeneity Moreover, the left-hand limit in hold, where (2.7) necessitates the sequence a to be bounded.
Proof. The inequalities relating Bohl exponents on different discrete intervals are evident from (2.2), as well as their homogeneity relations (2.5). Furthermore, define α := sup j∈I |a j | and for the sake of a convenient notation abbreviate Therefore, because the norm |·| is submultiplicative, j∈J a j ≤ j∈J |a j | ≤ α n for all J ∈ I n implies n √ φ n ≤ α for every n ∈ N and consequently β(a) ≤ α. (a) Let m, n ∈ N and suppose J ∈ I m+n denotes an arbitrary discrete interval, e.g. of the form J = [κ, κ + m) Z ∪ [κ + m, κ + m + n) Z with some κ ∈ I. Again the submultiplicativity of the norm allows us to obtain j∈J a j ≤ κ+m−1 j=κ a j κ+m+n−1 j=κ+m a j ≤ φ m φ n for all m, n ∈ N and since J ∈ I m+n was arbitrary, we can pass to the least upper bound over all such discrete intervals J yielding 0 ≤ φ m+n ≤ φ m φ n for all m, n ∈ N. Now it is well-known (see, e.g., [1, p. 246]) that the real sequence ( n √ φ n ) n∈N converges to the value inf n∈N n √ φ n , which establishes (2.6).
In order to deduce the characterization (2.7), we abbreviate the right-hand side of the inequality required in (2.7) by R. Thus, for every ε > 0 there exists a K ≥ 0 such that φ n ≤ K(R + ε) n for all n ∈ N and β(a) ≤ R follows from Conversely, it remains to show R ≤ β(a). From β(a) = lim ν→∞ sup n≥ν n √ φ n we see that for every sufficiently small ε > 0 there exists a N ∈ N such that the inequality sup n≥ν Combining this with (2.8), and since ε > 0 was arbitrary, we get R ≤ β(a). Proof. It is clear that both sequences a and |a| have the same Bohl exponents. When each a k ∈ A, k ∈ I, is invertible, then (2.9) follows from the proof of Prop. 2.2 applied toã k := a −1 k . As temporary conclusion, we present the close connection between the dichotomy spectrum and Bohl exponents of scalar difference eqns. (∆ a ): then (∆ a ) has the dichotomy spectrum Σ(a) = [β(a), β(a)].

Weighted shift operators.
For one-sided time I = Z + κ , κ ∈ Z, the dichotomy spectrum Σ + (A) of difference eqns. (∆ A ) and the essential (Fredholm) spectrum σ F of the unilateral matrix-weighted shift are related by (cf. [10] or [37,Thm. 3.22]) The set σ F (T A ) is rotationally invariant, i.e. consists of concentric rings and annuli in the complex plane. This observation has the striking advantage that information on the dichotomy spectrum can be obtained from results on shifts, like for instance Example 2.5 (asymptotically periodic scalar equations). Let p ∈ N and suppose (a k ) κ≤k is a sequence in K. If (|a k |) κ≤k is asymptotically p-periodic, i.e. there is some p-periodic positive real sequence (p k ) k∈I satisfying lim k→∞ (|a k | − p k ) = 0, then Ex. B.3 yields the dichotomy spectrum with the asymptotic mean c := p √ p κ+p−1 · · · p κ .
Difference eqns. (∆ A ) defined on the whole axis I = Z exhibit a richer spectral theory; it is based on bilateral matrix-weighted shifts As motivated in Sect. 1, it is advisable to distinguish different dichotomy spectra where Σ a (A) := Σ(A) denotes the dichotomy spectrum of (∆ A ), while its subspectra Σ s (A), Σ F (A), Σ F0 (A) and Σ π (A) are called surjectivity, Fredholm, Weyl resp. approximate point spectrum (see [37,39]). They consist of all reals γ > 0 such that fails to be onto, Fredholm, Weyl resp. bounded below. The corresponding spectra σ α are introduced in Sect. A.

Examples for
The upcoming examples feature sequences whose Lyapunov and Bohl exponents can be obtained explicitly. This equips us with a number of difference equations sufficiently flexible to illustrate our results later on. (1) If |a k | ≡ᾱ withᾱ > 0, then all Bohl and Lyapunov exponents coincide, i.e., One has discrete dichotomy spectra Σ α (a) = {ᾱ} for all α ∈ {a, F 0 , F, s, π}.
(2) If |a| is p-periodic, p ∈ N, then the Bohl and Lyapunov exponents become and we also arrive at the discrete spectra for all α ∈ {a, F 0 , F, s, π} and k ∈ Z.
(3) The both-sided asymptotically constant situation lim k→±∞ |a k | = α ± with reals α ± > 0 now illustrates a distinction between Bohl and Lyapunov exponents as well as their dependence on the discrete interval. Using [37, Ex. 5.3] one deduces To clarify that the inequalities (2.4) can be strict, for given reals α, β > 0 consider a sequence (a k ) k≥0 satisfying Hence, the modulus of a k is alternately equal to the constant values α resp. β on arithmetically increasing intervals (see Fig. 1). This yields the Bohl exponents In order to obtain the Lyapunov exponents of a, we observe that for every k ≥ 1 there exist unique n ∈ N and l ∈ [0, n) Z with k = n(n+1) 2 + l. For even n this implies and for odd n it is By means of these representations it is not difficult to deduce that the Lyapunov exponents are given as geometric mean λ + (a) = λ + (a) = √ αβ and fulfill the inequality β(a) ≤ λ + (a) = λ + (a) ≤ β(a); this corresponds with (2.4). Let us continue with a 2-dimensional problem from [39, Ex. 5.10] being useful in several situations: It illustrates both that the dichotomy spectrum is upper-semicontinuous, and that it might be smaller than the union of its diagonal spectra: satisfying A ∈ L ∞ (R 2 ) and involving the real sequences with reals α ± , β ± > 0 and a parameter λ ∈ R.
(1) Evidently, the background to obtain the Lyapunov spectra and filtrations are the transition matrices Φ(k, κ) of (∆ A ) and Φ * (k, κ) of (∆ * A ): • For α + = β + they are given by The explicit values for these quantities can be found in Tab. 1.
for all k, κ ≤ 0 and consequently yield The related dichotomy spectra Σ(A) for the various constellations of α ± = β ± were computed in [39, Ex. 5.10] already.
3. Spectra of diagonal equations. As a foretaste of and distinction from the general triangular situation, let us initially tackle a simpler case: Results on scalar difference eqns.
and bounded diagonal sequences (a 1 k ) k∈I , . . . , (a d k ) k∈I . Since the half line situation I = Z + κ was tackled in [37,Cor. 3.25] already, we restrict to the whole axis I = Z.
, the above Thm. 3.1 can also be understood as diagonal dominance criterion for an ED on Z. This means that the hyperbolicity assumption 1 ∈ Σ(a i ) for all 1 ≤ i ≤ d, implies that also the perturbed equation has an ED, provided sup k∈I |B k | is sufficiently small. In case the diagonal elements of each matrix B k vanish identically, this is actually related to the concept of diagonal dominance treated e.g. in [34] (for the lack of a reference in discrete time). Note that [34] states explicit conditions on the entries of B k guaranteeing an ED, whereas Thm. 3.1 only allows a qualitative statement.
Proof. For 1 ≤ i ≤ d we define the linear operators This section provides several criteria guaranteeing that a block-triangular difference equation (B) is diagonally significant, i.e. its dichotomy spectrum is given as union of the block-diagonal spectra. While this is not an issue on the half line (see Thm. 4.1), the full axis situation requires more care. As main results, we show that Σ(A) is always contained between the symmetric difference and the union of the block diagonal spectra Σ(A 1 ) and Σ(A 2 ) (for this, cf. Thm Proof. We represent points x ∈ K d as pairs (x 1 , x 2 ) with the components x i ∈ K di , i = 1, 2, and introduce the unilateral shift operator T A ∈ L( 2 ) as With the bounded projection P ∈ L( 2 ), (P φ) k := φ 1 k 0 and the closed subspaces X := R(P ), Y := N (P ) of 2 one obtains the direct sum 2 = X ⊕ Y and Furthermore, T A ∈ L( 2 ) can be represented as upper-triangular matrix operator With (2.11) in mind this yields the claim.

4.2.
Equations on the whole line I = Z. On the whole integer axis I = Z the statement of Thm. 4.1 is in general false and additional assumptions are required to obtain diagonal significance.
In the present context, diagonal significance of a block-triangular equation (B) w.r.t. a spectrum Σ α , α ∈ {a, F, F 0 , s, π}, means that holds. Indeed, the Ex. 2.7 (2) shows that the dichotomy spectrum Σ(A) of a blocktriangular eqn. (B) can be strictly smaller than the union Σ(A 1 ) ∪ Σ(A 2 ). However, this shrinking has no effect on the stability radius (see Cor. 4.4).
There are two approaches to determine subsets of Σ(A). The first one is based on well-known relations between the half line spectra and the spectra on Z: Proof. Thanks to [37,Cor. 4.30] has no interior points, then (DS α ) holds.
The following construction has prototype character for our investigations and closely resembles the proof of Thm. 4.1.
Proof. Let us represent x ∈ K d as pairs (x 1 , x 2 ) with the components x i ∈ K di for i = 1, 2. First, this allows us to introduce a bilateral shift T A ∈ L( 2 ), and second, P ∈ L( 2 ), (P φ) k := φ 1 k 0 defines a projection. Therefore, X := R(P ), Y := N (P ) are closed subspaces of 2 = X ⊕ Y and it holds Moreover, T A can be written as T A =  (2.12)). The second claimed inclusion anew follows using (2.12).
Rather than its whole line dichotomy spectrum Σ(A), the stability radius of (B) turns out to be fully determined by the diagonal blocks: and this implies the claim.
Proof. With the shifts T A , T A 1 and T A 2 defined in the proof of Thm. 4.3, we obtain from [45, (7)]. Because the intersection of both sides in this relation with R + distributes over the set operations involved, the claim results with (2.12).
Proof. By means of (A.1) this follows as above using [45,Cor. 3

.2] and (2.12).
A problem with the above criteria for diagonal significance is that certain dichotomy spectra, as well as their subspectra, have to be known in advance. In the following, we will thus obtain sufficient conditions on the basis of Lyapunov filtrations alone. As a further advantage, these criteria also provide diagonal significance of subspectra. To be precise, let us suppose that the diagonal systems (∆ A i ) and their adjoint eqns. (∆ * A i ) have Lyapunov spectra and filtrations λ i 1 , . . . , λ i ni , . Given this, one is able to formulate the following conditions on the Lyapunov spectra of the diagonal systems Note that we will illustrate these conditions in the Exs. 4.10 and 4.14 below.
Proof. We borrow our notation from the proof of Thm. 4.3. Using (B.3) we know that the adjoint shift T * A 1 has the SVEP if and only if holds. Let us first establish that this inequality is equivalent to (S * A 1 ): (⇒) The definition of the Lyapunov filtration for (∆ * A 1 ) guarantees χ * A 1 (x) = µ 1 j for x ∈ V 1 j \ V 1 j−1 , 1 ≤ j ≤ n * 1 and consequently (S * A 1 ) holds. (⇐) Conversely, assume (S * A 1 ) is satisfied and choose x ∈ K d \{0} arbitrarily. There exists a maximal 1 ≤ j ≤ n * 1 such that x ∈ V 1 j \ V 1 j−1 and thus χ * A 1 (x) = µ 1 j . Hence, and since x = 0 was arbitrary, T * A 1 has the SVEP. Analogously one uses (B.2) to show that (S A 2 ) is equivalent to the SVEP of T A 2 .
Proof. Thanks to (2.12) it again suffices to establish σ α (T A ) = σ α (T A 1 ) ∪ σ α (T A 2 ) with the shifts T A 1 , T A 2 defined in the proof of Thm. 4.3. Under the assumption (i) or (ii) this follows from [15, Lemma 2.2]. As in the above proof of Thm. 4.8 one shows that (iii) and (iv) are equivalent to the SVEP of T * A 1 resp. of T A 2 . Therefore, [15, Thm. 2.3] applies and yields the claim.
In order to obtain results on the diagonal significance of the Weyl dichotomy spectrum Σ F0 (A), one has to impose assumptions dual to (S * A 1 ) and (S A 2 ), namely Example 4.10. For the real sequences a 1 , a 2 from Ex. 2.7 the equivalences Asymptotically periodic sequences a as in Ex. 2.6(4) fulfill (S a ) or (S * a ) if and only if their asymptotic means c − , c + > 0 satisfy c − ≤ c + resp. c + ≤ c − .
. Proof. In the proof of Thm. 4.8 we have shown that (S * A 1 ) is equivalent to the SVEP of T * A 1 and that (S A 2 ) holds if and only if T A 2 has the SVEP. Along the same lines one verifies the equivalence of (S A 1 ) to the SVEP of T A 1 resp. that (S * A 2 ) is equivalent to a SVEP of T * A 2 . Given this, in a formal logical language our assumptions can be formulated as ((S * A 1 ) ∨ (S A 2 )) ∧ ((S A 1 ) ∨ (S * A 2 )), which is synonymous to the expression ). Hence, [45,Cors. 3.10 and 3.11] apply and yield σ F0 (T A ) = σ F0 (T A 1 ) ∪ σ F0 (T A 2 ). We intersect both sides of this equation with R + and from (2.12) one gets due to distributivity of the set relations, and thus the claim.
We close with several statements concerning the conditions (S A i ) and (S * A i ), which compare forward and backward growth of a difference equation (∆ A ) resp. its adjoint (∆ * A ), i ∈ {1, 2}. In the classical periodic situation they are fulfilled: , is discrete, due to (2.12) the spectrum σ(T A i ) consists of finitely many concentric circles. Then the inclusion ∂σ( Remark 4.13 (the classes P p (K d ) and P * p (K d )). In [39] we consider linear difference eqns. (∆ A ) with coefficient sequences in the classes which are related to the above assumptions. Indeed, by means of [39,Prop. A.3] one establishes the implications Example 4.14. We revisit the planar upper-triangular difference eqn. (∆ A ) from Ex. 2.7. The dichotomy spectra for its diagonal sequences are given in Ex. 2.6(3) and moreover [37,Ex. 5.3] yields the surjectivity dichotomy spectra (1) Since the Fredholm spectra Σ F (a i ) are discrete, one obtains from Thm because Cor. 4.7 applies for discrete subspectra Σ s (a 1 ), Σ π (a 2 ), i.e. (4.2).
(2) By means of Lyapunov exponent-like conditions we obtain the following criteria for diagonal significance. Analogously to the above Ex. 4.10 the condition • (S * a 1 ) is equivalent to α + ≤ α − and so Thm. 4.8(a) leads to (DS π ) • (S a 2 ) is equivalent to β − ≤ β + and Thm. 4

Conditions on C.
In order to provide sufficient conditions for diagonal significance on basis of the sequence C = (C k ) k∈Z alone, we define the linear spaces  Proof. We abbreviate 2 i := 2 (K di ) and for shifts T A i ∈ L( 2 i ), i = 1, 2, the generalized derivation ∆ : and thus ∆T C = 0, i.e. T C is in the kernel of ∆. Second, in case C ∈ R(A) with This yields ∆M X = T C and hence T C is in the range of ∆. By linearity we conclude that for elements C ∈ N (A) + R(A) the corresponding shifts T C are contained in the sum N (∆) + R(∆). Consequently, [7, Thm. 1] implies σ(T A ) = σ(T A 1 ) ∪ σ(T A 2 ) and the claim follows from (2.12).
We finally illuminate the close relation between the assumption of Thm. 4.15 and exponential dichotomies resp. trichotomies as discussed in [35]: (1) The linear space R(A) consists of all matrix sequences Y ∈ L ∞ (K d1 ) such that the matrix difference eqn. X k+1 = (A 1 k X k − Y k )(A 2 k ) −1 has a bounded solution. This, in turn, holds provided the linearly-homogenous equation has an exponential trichotomy on Z (cf. In this context, diagonal significance means that points of continuity of the dichotomy spectra Σ α (A i ) for the block subsystems guarantee that also the full system (B) has a corresponding spectrum being continuous at A.
Our analysis is fundamentally follows from the geometrically evident The remaining section is based on the assumption that A, A 1 and A 2 fulfill (2.1). Then, in our preparatory paper [39] we have shown that is a sufficient condition for Σ to be continuous at (∆ A ), while guarantees the corresponding continuity of the Weyl spectrum Σ F0 .

Triangular equations.
This section is concerned with linear eqns. (∆ A ) whose coefficient matrices A k are triangular, where w.l.o.g. we restrict to the upper-triangular situation. Hence, they are of the form with bounded diagonal sequences (b i k ) k∈I and bounded super-diagonal sequences (b i,j k ) k∈I for indices 1 ≤ i < j ≤ d in K.
As already pointed out, the importance of triangular equations (T ) is due to the close connection between the dichotomy spectrum and the Bohl exponents of their diagonal sequences. For the reader's convenience we therefore illustrate how an arbitrary linear equation (∆ A ) can be brought into triangular form: Algorithm 5.1. Choose an initial time κ ∈ I, e.g. κ := min I when the discrete interval I is bounded below. In case K = R: (0) Set T κ := id R d and determine the QR-decomposition and an upper-triangular B κ ∈ L(R d ) (see [21, p. 89, Thm. 2.1.14]) (1) For k > κ recursively compute the corresponding QR-decompositions with an orthogonal T k−1 and upper-triangular B k−1 In case K = C the same procedure applies, where T k ∈ L(C d ) is a sequence of unitary matrices and the transposes T T k have to replaced by the conjugate transposes T * k . By construction, the linear equation (∆ A ) is kinematically similar to the upper-triangular equation (T ) by means of the Lyapunov transformation (T k ) k∈I . Because of [37,Rem. 4.33] this has no effect on the dichotomy spectra Σ α .
On the half line I = Z + κ it is known that Σ + (B) is simply the union of spectra for the corresponding diagonal eqns. (∆ b i ) (see [37,Cor. 3.25]). As again demonstrated in Ex. 2.7(2), the situation for I = Z is more complicated.
So let us consider the whole line case I = Z from now on. In Thm. 5.2 we first provide a constructive tool to embed the dichotomy spectrum Σ(B) between a sub-and a superset. The "empty interior" condition from Thm. 4.3(b) extends to triangular equations (see Cor. 5.6), but we are able to complement it as follows: Rather than the Bohl exponents of the diagonal sequences, it suffices to assume appropriate inequalities for their Lyapunov exponents, in order to obtain diagonal significance (see Thm. 5.7).
Theorem 5.2. The dichotomy spectrum Σ(B) of (T ) satisfies with a set Σ 1 given by and for d > 2 allowing the recursive construction Here, the square matrix B j k ∈ K (d−j)×(d−j) is defined by simultaneously discarding the first j ∈ [0, d) Z columns and rows of B k ; it is B k = B 0 k . Setting Σ j := Σ(A j ) and applying Thm. 4.3(a) to (5.1) we deduce The above Thm. 5.2 allows to circumscribe the dichotomy spectrum using exclusively the diagonal spectral intervals. As a concrete example we consider and bounded super-diagonal entries, the following holds: In the terminology of Thm. 5.2 with d = 4 we obtain the inclusions [3,4] which guarantee Σ(b 2 ) ∩ Σ 2 = {3}, consequently [1,4] ⊆ Σ(b 2 ) Σ 2 and thus the inclusions [1,4] Yet, one obtains the stability radius of (T ) from the diagonal sequences: According to Ex. 2.7(2) one knows that Cor. 5.6 is wrong without the additional assumption on interior points. Proof. Retaining the notation from the proof of Thm. 5.2, set σ j := Σ(b j ). By assumption, the intersection and if we invest our assumption, σ j ∩ Σ j has no interior points. Hence, Thm.
The next criteria for diagonal significance involve only Lyapunov exponents of the first resp. last d − 1 diagonal sequences. With Ex. 2.6(5) one easily constructs diagonally significant equations having overlapping diagonal spectral intervals.
Proof. First of all, using Lemma B.5 we obtain from assumption (i) that every T b i has the SVEP for 1 < i ≤ d, while assumption (ii) guarantees the SVEP of T * b i for all 1 ≤ i < d. Because mathematical induction on basis of [15,Thm. 2.3] implies the relation σ α (T A ) = d i=1 σ α (T b i ), the claim results from (2.12).  In order to release this paper from its somehow theoretical flavor, let us consider the nonlinear planar difference equation

In both cases one has
Concerning the relevance of (6.1) in population dynamics we refer to [12], where it is known as Leslie-Gower equation. The real nonnegative sequences (a k ) k∈Z , (b k ) k∈Z , (c k ) k∈Z and (d k ) k∈Z are assumed to be bounded.
In the extinction equilibrium (0, 0) the variational equation of (6.1) is of diagonal form (D) with 0 for all k ∈ Z of (6.1), which remains on the x-axis. Along this solution, the variational equation has the coefficients and we distinguish two cases: (a) For a sequence a k ≡ λ > 1 one obtains x * k ≡ λ − 1 and A k = ; hence one spectral interval is a singleton 1 λ . Then Thm. 4.3(b) implies diagonal significance Σ(A) = 1 λ ∪Σ( c 1+(1−λ)d ) and the dependence of Σ(A) on λ is shown in Fig. 2. It indicates uniform asymptotic stability of the constant extinction solution (λ − 1, 0) (at least) for λ ≥ 2, while it is a nonautonomous saddle for λ > 1 near 1. (b) Allowing the sequence a to vary and considering c k ≡ λ > 0 as parameter, the solution x * is not constant anymore. Thus, the spectra of the diagonal sequences can be closed intervals (of positive length, see Fig. 3). As long as  7. Outlook and remarks on the continuous time situation. As both a conclusion and outlook let us point out that our results are useful in an ODE context as well: Thereto, consider a linear ODĖ with e.g. a locally integrable coefficient matrix A : R → L(K d ) and the transition matrix U (t, s) ∈ L(K d ).
On the one hand, the discrete time results of this paper might be applied to the time-1-difference equation (∆ A ) with the invertible coefficients A k := U (k + 1, k) for all k ∈ Z; they are bounded under the assumption sup k∈Z |U (k + 1, k)| < ∞. It is not difficult to see that a (block-)triangular structure of (7.1) extends to the matrices A k . Furthermore, we restricted ourselves to linear difference equations with invertible coefficient matrices. This excludes various features present in discrete but not in continuous time. Thus, as long as the dichotomy spectrum is concerned, similar results as in Sects. 3-5 are expected to hold for (block-)triangular ODEs (7.1). Yet, [9] specifically treats ODEs (7.1) and provides not only necessary and sufficient conditions for diagonal significance in the dichotomy spectrum, but also a procedure to obtain Σ(A) from the half line spectra.
On the other hand, the subspectra Σ s , Σ F and Σ F0 were introduced in order to classify nonautonomous bifurcations (cf. [37]). They appear equally important for (7.1) and nonautonomous nonlinear ODEs. Yet, their corresponding continuous time theory has not been established so far.
Appendix A. Operators on Hilbert spaces. Let X be an infinite-dimensional separable and complex Hilbert space with inner product ·, · . The set of linear bounded operators between X and a normed space Y is abbreviated as L(X, Y ); we write N (S) ⊆ X for the kernel and R(S) ⊆ Y for the range of some S ∈ L(X, Y ). Furthermore, L(X) := L(X, X) is the Banach algebra of bounded linear operators on X with identity id X .
Given an operator T ∈ L(X), let σ a (T ) := σ(T ), σ π (T ), σ s (T ), σ F (T ) and σ F0 (T ) be its spectrum, approximate point spectrum, surjectivity, essential and Weyl spectrum, respectively (see [1,2,3,29]). Since X is a Hilbert space, the left spectrum σ l (T ) resp. the right spectrum σ r (T ) satisfy where Ω * := λ ∈ C : λ ∈ Ω for every Ω ⊆ C. An operator T ∈ L(X) possesses the single-valued extension property (SVEP for short) at a point λ 0 , provided for every neighborhood U ⊆ C of λ 0 the only analytic function f : U → X satisfying (λ id X −T )f (λ) ≡ 0 on U is identically vanishing. If the SVEP holds for every λ 0 ∈ C, then the operator T is said to have the SVEP. The associate set (cf. [3, p.  Appendix B. Weighted shift operators. Let I be a discrete interval unbounded above. We denote by 2 the linear space of square-summable sequences φ = (φ k ) k∈I in K d equipped with the inner product φ, ψ := k∈I φ k , ψ k for all φ, ψ ∈ 2 and norm φ = φ, φ ; 2 is the prototype of a separable Hilbert space. For a bounded weight sequence A = (A k ) k∈I in L(K d ), we define the left shift Lemma B.1. The adjoint of T A is given by T * A ∈ L( 2 ), (T * A φ) k = A * k φ k+1 , k ∈ I. Proof. For arbitrary φ, ψ ∈ 2 we obtain T A φ, ψ = k∈I A k φ k , ψ k+1 = k∈I φ k , A * k ψ k+1 = φ, T * A ψ with (T * A ψ) k := A * k ψ k+1 for all k ∈ I.
B.2. Bilateral shifts. For I = Z one speaks of bilateral shifts T A ∈ L( 2 ). In case of invertible weights A k ∈ K d×d the condition sup k∈Z A −1 k < ∞ implies 0 ∈ σ(T A ). As opposed to unilateral shifts, a characterization of the SVEP is more involved: • T A has the SVEP if and only if (cf. [ for some κ ∈ Z. Different from the scalar situation (cf. Lemma B.5 below), both T A and T * A might fail to possess the SVEP (see [11,Ex. 2.3]). Particularly for scalar shifts with weights a ∈ ∞ (K), thanks to [40,Thm. 3]  Example B.4 (asymptotically periodic case). Let κ ∈ Z, p + , p − ∈ N, and (a k ) k∈Z be a sequence in K. If (|a k |) k≥κ is asymptotically p + -periodic and (|a k |) k≤κ is asymptotically p − -periodic with asymptotic means c + , c − , then [40,Thm. 5] showed