Lower spectral radius and spectral mapping theorem for suprema preserving mappings

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max type kernel operators.


1.
Introduction. Max-type operators (and corresponding max-plus type operators and their tropical versions known also as Bellman operators) arise in a large field of problems from the theory of differential and difference equations, mathematical physics, optimal control problems, discrete mathematics, turnpike theory, mathematical economics, mathematical biology, games and controlled Markov processes, generalized solutions of the Hamilton-Jacobi-Bellman differential equations, continuously observed and controlled quantum systems, discrete and continuous dynamical systems, ... (see e.g. [28], [20], [27], [26], [4], [32] and the references cited there). The eigenproblem of such operators has so far received substantial attention due to its applicability in the above mentioned problems (see e.g. [28], [20], [4], [3], [32], [22], [29], [13], [31], [30], [2], [41] and the references cited there). However, there seems to be a lack of general treatment of spectral theory for such operators, even though the spectral theory for nonlinear operators on Banach spaces is already quite well developed (see e.g. [11], [10], [12], [16], [40], [32] and the references cited there). One of the reasons for this might lie in the fact that these operators behave nicely on a suitable subcone (or subsemimodule), but less nicely on the whole (Banach) space. Therefore it appears, that it is not trivial to directly apply this known non-linear spectral theory to obtain satisfactory information on a restriction to a given cone of a max-type operator. The Bonsall cone spectral radius plays the role of the spectral radius in this theory (see e.g. [28], [29], [32], [4], [21], [17], [31] and the references cited there).
In [32], we studied Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We showed that for such a mapping the Bonsall cone spectral radius is the maximum value of its approximate point spectrum (see Theorem 2.1 below). We also proved that an analogue of this result holds also for Lipschitz, positively homogeneous and additive mappings defined on a normal convex cone in a normed space (see Theorem 2.2 below).
The current article may be considered as a continuation of [32]. It is organized as follows. In Section 2 we recall some definitions and results that we will use in the sequel. We show in Section 3 that the lower spectral radius of a mapping from both of the above decribed settings is a minimum value of its approximate point spectrum (Theorems 3.5 and 3.6). In Section 4 we apply this result to show that the maxpolynomial spectral mapping theorem holds for the approximate point spectrum of Lipschitz, positively homogeneous and finite suprema preserving mappings (Theorem 4.11). In the last section we use this spectral mapping theorem to prove some new inequalities for the Bonsall cone spectral radius of Hadamard products of max type kernel operators (Theorem 5.5).

Preliminaries.
A subset C of a real vector space X is called a cone (with vertex 0) if tC ⊂ C for all t ≥ 0, where tC = {tx : x ∈ C}. A map A : C → C is called positively homogeneous (of degree 1) if A(tx) = tA(x) for all t ≥ 0 and x ∈ C. We say that the cone C is pointed if C ∩ (−C) = {0}.
A convex pointed cone C of X induces on X a partial ordering ≤, which is defined by x ≤ y if and only if y − x ∈ C. In this case C is denoted by X + and X is called an ordered vector space. If, in addition, X is a normed space then it is called an ordered normed space. If, in addition, the norm is complete, then X is called an ordered Banach space.
A convex cone C of X is called a wedge. A wedge induces on X (by the above relation) a vector preordering ≤ (which is reflexive, transitive, but not necessary antisymmetric). We say that the cone C is proper if it is closed, convex and pointed. A cone C of a normed space X is called normal if there exists a constant M such that x ≤ M y whenever x ≤ y, x, y ∈ C. A convex and pointed cone C = X + of an ordered normed space X is normal if and only if there exists an equivalent monotone norm ||| · ||| on X, i.e., |||x||| ≤ |||y||| whenever 0 ≤ x ≤ y (see e.g. [9,Theorem 2.38]). Every proper cone C in a finite dimensional Banach space is necessarily normal.
If X is a normed linear space, then a cone C in X is said to be complete if it is a complete metric space in the topology induced by X. In the case when X is a Banach space this is equivalent to C being closed in X.
If X is an ordered vector space, then a cone C ⊂ X + is called a max-cone if for every pair x, y ∈ C there exists a supremum x ∨ y (least upper bound) in C. We consider here on C an order inherited from X + . A map A : C → C preserves finite suprema on C if A(x ∨ y) = Ax ∨ Ay (x, y ∈ C). If A : C → C preserves finite suprema, then it is monotone (order preserving) on C, i.e., Ax ≤ Ay whenever x ≤ y, x, y ∈ C .
An ordered vector space X is called a vector lattice (or a Riesz space) if every two vectors x, y ∈ X have a supremum and infimum (greatest lower bound) in X. A positive cone X + of a vector lattice X is called a lattice cone.
Note that by [9, Corollary 1.18] a pointed convex cone C = X + of an ordered vector space X is the lattice cone for the vector subspace C − C generated by C in X, if and only if C is a max cone (in this case a supremum of x, y ∈ C exists in C if only if it exists in X; and suprema coincide).
If X is a vector lattice, then the absolute value of x ∈ X is defined by |x| = x ∨ (−x). A vector lattice and normed vector space is called a normed vector lattice (a normed Riesz space) if |x| ≤ |y| implies x ≤ y . A complete normed vector lattice is called a Banach lattice. A positive cone X + of a normed vector lattice X is proper and normal.
In a vector lattice X the following Birkhoff's inequality for x 1 , . . . , x n , y 1 , . . . , y n ∈ X holds: For the theory of vector lattices, Banach lattices, cones, wedges, operators on cones and applications e.g. in financial mathematics we refer the reader to [1], [9], [7], [42], [6], [23], [18], [32], [24], [5] and the references cited there. Let X be a normed space and C ⊂ X a non-zero cone. Let A : C → C be positively homogeneous and bounded, i.e., It is easy to see that A = sup{ Ax : x ∈ C, x ≤ 1} and A m+n ≤ A m · A n for all m, n ∈ N. It is well known that this implies that the limit lim n→∞ A n 1/n exists and is equal to inf n A n 1/n . The limit r(A) := lim n→∞ A n 1/n is called the Bonsall cone spectral radius of A. The approximate point spectrum σ ap (A) of A is defined as the set of all s ≥ 0 such that inf{ Ax − sx : x ∈ C, x = 1} = 0. The (distinguished) point spectrum σ p (A) of A is defined by For x ∈ C define the local cone spectral radius by r x (A) := lim sup n→∞ A n x 1/n . Clearly r x (A) ≤ r(A) for all x ∈ C. It is known that the equality is not valid in general. In [28] there is an example of a proper cone C in a Banach space X and a positively homogeneous and continuous (hence bounded) map A : If X is a Banach lattice, C ⊂ X + a max-cone and A : C → C a mapping which is bounded, positively homogeneous and preserves finite suprema, then the equality (2) is not necessary valid as shown in [32]. Some additional examples of maps for which (2) is not valid can be found in [17].
Let C be a cone in a normed space X and A : C → C. Then A is called Lipschitz if there exists L > 0 such that Ax − Ay ≤ L x − y for all x, y ∈ C.
The following two results were the main results of [32] (see [ Theorem 2.1. Let X be a normed vector lattice, let C ⊂ X + be a non-zero maxcone. Let A : C → C be a mapping which is bounded, positively homogeneous and preserves finite suprema. Let C ′ ⊂ C be a bounded subset satisfying A n = sup{ A n x : x ∈ C ′ } for all n. Then If, in addition, A is a Lipschitz, then r(A) = max t : t ∈ σ ap (A) .
Theorem 2.2. Let X be a normed space, C ⊂ X a non-zero normal wedge and let A : C → C be positively homogeneous, additive and Lipschitz. Let Then In particular, 3. Lower spectral radius and approximate point spectrum. Let X be a normed space and C ⊂ X a non-zero cone. Let First we observe the following result.
Proposition 3.1. Let X be a normed space and C ⊂ X a non-zero cone. If A : C → C is positively homogeneous and bounded, then If, in addition, A is Lipschitz and t ∈ σ ap (A) then one can see easily that m(A) ≤ t ≤ A and t n ∈ σ ap (A n ) for all n ∈ N (see also the proof of [32, Lemma 3.3]). Since m(A n ) ≤ t n ≤ A n , it follows that d(A) ≤ t ≤ r(A), which completes the proof.

LOWER RADIUS AND SPECTRAL MAPPING THEOREM 5
The following example shows that σ ap (A) may not contain the whole interval Example 3.2. Let X = ℓ ∞ with the standard basis e n,k (n, k ∈ N). More precisely, the elements of X are formal sums x = n,k∈N α n,k e n,k with real coefficient α n,k such that x := sup{|α n,k | : n, k ∈ N} < ∞. Then X is a Banach lattice with the natural order. Let C = X + and let A : C → C be defined by Ae n,1 = n −1 e n,2 , Ae n,k = e n,k+1 (k ≥ 2). More precisely, By Theorem 2.1, r(A) = max{t : t ∈ σ ap (A)} if X is a normed vector lattice, C ⊂ X + a non-zero max-cone and T : C → C a mapping, which is Lipschitz, positively homogeneous and preserves finite suprema. We show below in Theorem 3.5 that under these assumptions we also have that d(A) = min{t : t ∈ σ ap (A)}. In the proof we will need the following lemmas ([32, Lemma 3.1, Lemma 3.2]). The first one is based on the inequality (1). x j − y j . Lemma 3.4. Let X be a vector lattice and x j , y j ∈ X for j = 1, . . . , n. Then If, in addition, X is a normed vector lattice and x j ≥ y j ≥ 0 for j = 1, . . . , n, then The following result is one of the main results of this section. Let ε > 0. We show that there exists w ∈ C, w = 0 such that Aw−w w ≤ ε. Let n ∈ N satisfy n > max{2, 16ε −2 } and m(A n ) > 0. Find δ > 0 such that Suppose on the contrary that 4a m < max{a m−1 , a m+1 } for all m = 1, . . . , N/n − 1.
If a j+1 ≥ a j for some j ≤ N n − 2, then this condition for j + 1 means that a j+2 > 4a j+1 . By induction we get Continuation of the Proof of Theorem 3.5. Let m, 1 ≤ m < N/n satisfy A mn x ≥ then Ay−y y ≤ ε and we are done. So assume that y < 8ε −1 A mn x . Let We have u ≥ n−1 n A mn x ≥ 1 2 A mn x . Furthermore, by Lemma 3.4 Since ε > 0 was arbitrary, 1 ∈ σ ap (A).
If, in addition, A is Lipschitz, then d(A) = min{t : t ∈ σ ap (A)} by Proposition 3.1.
By replacing ∨ with + in the proof above and by suitably adjusting some estimates the following theorem also follows. We include some details of the proof for the sake of clarity. If, in addition, A is Lipschitz, then d(A) = min t : t ∈ σ ap (A) .
Proof. As in the proof of Theorem 3.5 we may assume that d(A) = 1. Let ε > 0 and let n ∈ N satisfy n > max{2, 32M ε −2 } and m(A n ) > 0, where M is the normality constant of C. If N ∈ N is chosen as in the proof of Theorem 3.5, then it follows in the same way that there exists m ∈ N, 1 ≤ m < N/n such that Let y = (m+1)n−1 j=(m−1)n A j x. Then Ay − y = A (m+1)n x − A (m−1)n x and so Ay − y ≤ 8 A mn x . Without loss of generality we may assume that y < 8ε −1 A mn x . Let Then 2 u ≥ A mn x and Since ε > 0 was arbitrary, 1 ∈ σ ap (A).
There are interesting non-trivial examples of operators to which Theorem 3.6 applies. In particular, it applies to the C-linear Perron-Frobenius operators from [29, Section 5] and [34, Sections 5 and 6].
Remark 3.7. In [32, Example 2] there is an example of a Banach lattice X and a closed max-cone C ⊂ X + , which is normal and convex (a normal wedge), and a positively homogeneous, additive mapping A : C → C that preserves all suprema and satisfies A = 1. However, A is not Lipschitz, d(A) = r(A) = 0 and {0, 1} ⊂ σ ap (A). So, as pointed out in [32], the Lipschitzity of A is necessary for the property r(A) = max{t : t ∈ σ ap (A)} to hold.

VLADIMIR MÜLLER AND ALJOŠA PEPERKO
The following example shows that the Lipschitzity of A is necessary for the property d(A) = min{t : t ∈ σ ap (A)} to hold in Theorems 3.5 and 3.6.
Then C is a max-cone (moreover C is a convex normal cone). Let A : C → C be defined by A n,k∈N α n,k e n,k = n∈N α n,1 e n,1 + n −1 α n,1 e n,2 + 2nα n,2 e n,3 + ∞ k=3 2α n,k e n,k+1 .
Clearly A is bounded, A = 2 and A is a positively homogeneous, additive mapping on C that preserves finite suprema. We have Ae n,1 − e n,1 = n −1 e n,2 = n −1 and A 2 e n,1 − Ae n,1 = 2e n,3 = 2 for all n, so 1 ∈ σ ap (A) and A is not Lipschitz. On the other hand, it is easy to see that d(A) = 2.
4. Spectral mapping theorems for point and approximate point spectrum. In this section we generalize the spectral mapping theorem in max-algebra (see [31,Theorem 3.4] and also [19,Theorem 3.6] ) to the infinite dimensional setting. The main results of this section are Corollary 4.3, Theorem 4.6 and Theorem 4.11.
Let X be a Riesz space (i.e., a vector lattice). Let C ⊂ X + be a nonzero max-cone and A : C → C a positively homogeneous mapping that preserves finite suprema. Let P + denote the set of all polynomials q(z) = n j=0 α j z j ∈ P + with α j ≥ 0 for all j. For q ∈ P + and t ≥ 0 let q ∨ (t) = max j α j t j (i.e., q ∨ is the maxpolynomial corresponding to q). Define q ∨ (A) : C → C by Lemma 4.1. Let X be a vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a positively homogeneous mapping that preserves finite suprema and q = n j=0 α j z j ∈ P + . Then q ∨ (σ p (A)) ⊂ σ p (q ∨ (A)).
Proof. Let t ≥ 0, x ∈ C and Ax = tx. Then A j x = t j x for all j ≥ 0, and so Lemma 4.2. Let X be a vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a positively homogeneous, finite suprema preserving mapping. Assume that q(z) = n j=1 α j z j ∈ P + , q ∨ (1) = 1 and 1 ∈ σ p (q ∨ (A)). Then 1 ∈ σ p (A).
Let x ∈ C be a nonzero vector satisfying q ∨ (A)x = x. Set y = m−1 Hence Ay = y and 1 ∈ σ p (A). Corollary 4.3. Let X be a vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a positively homogeneous, finite suprema preserving mapping and q = n j=1 α j z j ∈ P + a non-zero polynomial. Then σ p (q ∨ (A)) = q ∨ (σ p (A)).
Lemma 4.4. Let X be a vector lattice, x, y ∈ X + , s > 1 and x ∨ y = sx. Then y = sx.
We have x + y ≥ x ∨ y = sx, and so y ≥ (s − 1)x. Let 1 ≤ j ≤ k − 1 and suppose that y ≥ j(s − 1)x. Since x ∨ y = sx, we have and so By the induction assumption for j = 1 applied to x ∨ y ′ = s ′ x, where y ′ = y−j(s−1)x

Now the following result follows.
Theorem 4.6. Let X be a vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a positively homogeneous, finite suprema preserving mapping and let q(z) = n j=0 α j z j ∈ P + . Then Proof. The first inclusion follows from Lemma 4.1.
Remark 4.7. Under the assumptions of Theorem 4.6 it is possible for q(z) = n j=0 α j z j that σ p (q ∨ (A)) = q ∨ (σ p (A)). Consider the Banach lattice ℓ ∞ with natural order and let C be the positive cone. Let (e n ) be the standard basis in ℓ ∞ and define a mapping A : C → C by A( n γ n e n ) = n n −1 γ n e n+1 . Then σ p (A) = ∅. Let q ∨ (z) = 1 ∨ z. Then for y = n e n we have q ∨ (A)y = y ∨ Ay = y.
In the following, X will be a normed vector lattice and C ⊂ X + a non-zero max cone. Let A : C → C be positive homogeneous, Lipschitz and finite suprema preserving. The spectral mapping theorem for the approximate point spectrum (see Theorem 4.11 below) can be proved similarly as the above results by repeating similar arguments. However, one can also apply the following standard construction.
Denote by ℓ ∞ (X) the set of all bounded sequences (x j ) ∞ j=1 of elements of X. With the norm (x j ) = sup j x j and order (x j ) ≤ (y j ) ⇔ x j ≤ y j for all j, ℓ ∞ (X) is again a normed vector lattice. Let C ∞ ⊂ ℓ ∞ (X) be the set of all bounded sequences of elements of C and let A ∞ : C ∞ → C ∞ be defined by A ∞ ((c j )) = (Ac j ).
Let c 0 (X) be the set of all null sequences (x j ) of elements of X, x j → 0. Clearly c 0 (X) is an ideal in ℓ ∞ (X). Let X = ℓ ∞ (X)/c 0 (X). Then X is again a normed lattice. Let C = (C ∞ + c 0 (X))/c 0 (X) and A : C → C be the natural quotient mapping, which is well defined since A is Lipschitz. Then C is a max cone and A is positive homogeneous and finite suprema preserving. Moreover, it is easy to show that σ ap (A) = σ p ( A).
Thus the following result follows.
Theorem 4.8. Let X be a normed vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a Lipschitz, positively homogeneous, finite suprema preserving mapping. Let q(z) = n j=0 α j z j ∈ P + . Then Corollary 4.9. Let X be a normed vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a Lipschitz, positively homogeneous, finite suprema preserving mapping. If q ∈ P + , q = deg q j=0 α j z j , then r(q ∨ (A)) = q ∨ (r(A)).
The equality (6) suggests that the situation for the approximate point spectrum is even better. Indeed, the equality σ ap (q ∨ (A)) = q ∨ (σ ap (A)) is true for all polynomials from P + as we prove below in Theorem 4.11. Proposition 4.10. Let X be a normed vector lattice and let C ⊂ X + be a nonzero max-cone. Let A : C → C be a Lipschitz, positively homogeneous, finite suprema preserving mapping. Let q(z) = α + z ∈ P + . Then σ ap (q ∨ (A)) = q ∨ (σ ap (A)).
Proof. If α = 0 then the statement is clear.
In particular, the results above apply to the following two classes of examples from [28] and [32].
On the other hand, We also point out the following related example from [32]. In particular, if M is the set of all natural numbers N, our results apply to infinite bounded non-negative matrices k = [k(i, j)] (i.e., k(i, j) ≥ 0 for all i, j ∈ N and k ∞ = sup i,j∈N k(i, j) < ∞). In this case, X = l ∞ and C = l ∞ + and A = k ∞ .
5. Application to inequalities involving Hadamard products. Throughout this section let X, C and all the mappings A, B, A 1 , . . . , A m , A 11 , . . . , A mn that map C to C be as in Example 4.12 (where the functions α and β are fixed -the same for all operators A, B, A 1 , . . . , A m , A 11 , . . . , A mn ) or let X, C and all the mappings A, B, A 1 , . . . , A m , A 11 , . . . , A mn that map C to C be as in Example 4.13. We denote the set of such mappings by C.
In this section we apply (6) to prove some new inequalities on Hadamard products (Theorem 5.5) by applying an idea from [15] . Let A • B denote the Hadamard (or Schur) product of mappings A and B from C, i.e., A • B ∈ C is a mapping with a kernel k(s, t)h(s, t), where k and h are the kernels of A and B, respectively. Similarly, for γ > 0 let A (γ) denote the Hadamard (or Schur) power of A, i.e., a mapping with a kernel k γ (s, t).
The following result was stated in [37,Theorem 4.1] in the special case of n × n non-negative matrices and was essentially proved in [36]. It follows from [36, Theorem 5.1 and Remark 5.2] and the fact that for A 1 , . . . , A m , A ∈ C and γ > 0 we have A (γ) 1 · · · A (γ) m = (A 1 · · · A m ) (γ) and A (γ) = A γ and consequently r(A (γ) ) = r(A) γ . Observe also that A ≤ B implies r(A) ≤ r(B).
Remark 5.2. An analogue to (7) for · also holds. As pointed out in [36], [37] and [30], Theorem 5.1 is in fact a result on the generalized and joint spectral radius in max-algebra (see also [38]). The logarithm of the latter is also known as the maximal Lyapunov exponent in max algebra and is important in the study of certain discrete event systems (see e.g. the references cited in [30]).
Inequalities (10) and (11) below were established in [37,Corollary 4.8] in the special case of n × n matrices, while the inequality (9) is a max-algebra version of [39,Corollary 3.3] and [15,Theorem 3.2]. The proofs of these inequalities are similar to the proofs of the results from [37], [15] and [39] and are included for the convenience of readers. In the proof we use the fact that r(AB) = r(BA).
Remark 5.4. As pointed out in [37,Example 4.10] the inequalities in (9) are sharp and may be strict, and in some cases the inequality (11) may be better than (10).
If m ∈ N and q ∈ P + , q = deg q j=0 α j z j let us define the polynomial q [m] by q [m] = deg q j=0 α m j z j . By applying (6) and an idea from [15], we extend Corollary 5.3 in the following way.