ON SOME BANACH SPACE CONSTANTS ARISING IN NONLINEAR FIXED POINT AND EIGENVALUE THEORY

As is well known

Although the above proof is complete, we still sketch another three implications.  Let e be a fixed point of f which exists by (a). If g(e) + e ≤ 1, then g(e) = 0, contradicting our assumption that g(B(X)) ⊆ X \ {0}. So, we must have g(e) + e > 1, hence e ∈ S(X) and g(e) = λe with λ = g(e) + e − 1 > 0. It is a striking fact that all four assertions of Theorem 1.1 are true if dimX < ∞, but false if dimX = ∞. This means that in any infinite-dimensional Banach space one may find not only fixed point free self-maps of the unit ball, but also retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. The first examples of this type have been constructed in special spaces; for the reader's ease we recall two of them, the first one due to Kakutani [22] and the second is due to Leray [24]. It is easy to see that f (x) = x for any x ∈ B( 2 ). By (1.5), this map gives rise to the operator which clearly has no positive eigenvalues (actually, no eigenvalues at all) on S( 2 ). (1.11) Then, the homotopy h :

Lipschitz conditions and measures of noncompactness
Given two metric spaces M and N and some (in general, nonlinear) operator F : M → N, we denote by Recall that a nonnegative set function φ defined on the bounded subsets of a normed space X is called measure of noncompactness if it satisfies the following requirements (A,B ⊂ X bounded, K ⊂ X compact, λ > 0): We point out that in the literature it is usually required that φ(coA) = φ(A), that is, φ is invariant with respect to the convex closure of a set A; however, since in our calculations we only need to consider convex closures of sets of the form A ∪ {0}, absorption invariance suffices for our purposes. The most important examples are the Kuratowski measure of noncompactness (or set measure of noncompactness) α(M) = inf{ε > 0 : M may be covered by finitely many sets of diameter ≤ ε}, (2.2) the Istrȃţescu measure of noncompactness (or lattice measure of noncompactness) is called the φ-norm of F. It follows directly from the definitions that φ(F) ≤ Lip(F) in case φ = α or φ = β. Moreover, if L is linear, then clearly Lip(L) = L , and so α(L) ≤ L and β(L) ≤ L . A detailed account of the theory and applications of measures of noncompactness may be found in the monographs [1,2].
In view of conditions (a) and (b) of Theorem 1.1, the two characteristics have found a considerable interest in the literature; we call (2.7) the Lipschitz constant and (2.8) the retraction constant of the space X. Surprisingly, for the characteristic (2.7), one has L(X) = 1 in each infinite-dimensional Banach space X. Clearly, L(X) ≥ 1, by the classical Banach-Caccioppoli fixed point theorem. On the other hand, it was proved in [26] satisfies Lip( f ) > 1, without loss of generality, then following [8] we fix ε ∈ (0,Lip( f ) − 1) and consider the map f ε : A straightforward computation shows then that every fixed point of f ε is also a fixed point of f , and that Lip( f ε ) ≤ 1 + ε, hence L(X) ≤ 1 + ε. On the other hand, calculating or estimating the characteristic (2.8) is highly nontrivial and requires rather sophisticated individual constructions in each space X (see [3,4,5,6,7,11,13,16,17,19,23,25,28,29,30,35] are known; a survey of such estimates and related problems may be found in the book [19] or, more recently, in [18]. In view of Theorem 1.1, it seems interesting to introduce yet another two characteristics, namely, which we call the eigenvalue constant of X, and Observe that, similarly as for the constant (2.7), the calculation of (2.11) is trivial, because E(X) = 0 in every infinite-dimensional space X. In fact, according to [26] we may choose first some fixed point free Lipschitz map f : B(X) → B(X), and then define a Lipschitz continuous map g : B(X) → X \ {0} without positive eigenvalues on S(X) as in (1.5). This shows that E(X) < ∞. Now, it suffices to observe that the eigenvalue equation g(e) = λe is invariant under rescaling, that is, the map εg has, for any ε > 0, no positive eigenvalues on S(X). But Lip(εg) = ε Lip(g), and so E(X) may be made arbitrarily small. If we define a homotopy h through a given Lipschitz continuous retraction ρ : B(X) → S(X) like in (1.6), then an easy calculation shows that (2.13) holds for h with k = Lip(ρ), and so H(X) ≤ R(X).
The main problem we are now interested in consists in finding (possibly sharp) estimates for φ(F), where F is one of the maps f , ρ, h, and g arising in Theorem 1.1, and φ is some measure of noncompactness (e.g., φ ∈ {α, β,γ}). To this end, for a normed space X we introduce the characteristics We point out that the paper [32] is concerned with characterizing some classes of spaces X in which the infimum L φ (X) = 1 is actually attained, that is, there exists a fixed point free φ-nonexpansive self-map of B(X). This is a nontrivial problem to which we will come back later (see the remarks after Theorem 3.3).
Theorem 3.1. The relations Proof. The fact that L φ (X) = 1 and E φ (X) = 0 is a trivial consequence of the estimate φ(F) ≤ Lip(F) and our discussion above. The proof of the implication (a)⇒(b) in Later (see Theorem 4.2), we will discuss a class of spaces in which the estimate in (3.1) also turns into equality.
The equality E(X) = 0 which we have obtained before for the characteristic (2.11) shows that in every Banach space X one may find "arbitrarily small" operators without zeros on B(X) and positive eigenvalues on S(X). Observe, however, that the infimum in (2.11) is not a minimum, since Lip(g) = 0 means that g is constant, say g(x) ≡ y 0 = 0, and then g has the positive eigenvalue λ = y 0 with normalized eigenvector e = y 0 / y 0 .
On the other hand, the equality E φ (X) = 0 for the characteristic (2.18) shows that in every Banach space X, one may find such operators which are "arbitrarily close to being compact". As we will show later (see Theorem 3.3), in this case the infimum in (2.18) is a minimum, that is, the operator g may always be chosen as a compact map. The operator g from (1.10) is not optimal in this sense, since g(e k ) = e k+1 − e k , where (e k ) k is the canonical basis in 2 , and thus φ(g) ≥ 1. In the following Example 3.2, we give a compact operator in 2 without positive eigenvalues. This example has been our motivation for proving the general result contained in the subsequent Theorem 3.3.
Recall that, given M ⊆ X, an operator F : M → Y , and a measure of noncompactness φ on X and Y , the characteristic is called the lower φ-norm of F. This characteristic is closely related to properness. In fact, from φ(F) > 0 it obviously follows that F is proper on closed bounded sets, that is, the preimage F −1 (N) of any compact set N ⊂ Y is compact. The converse is not true: for example, the operator F : X → X defined on an infinite-dimensional space X by F(x) := x x is a homeomorphism with inverse F −1 (y) = y/ y for y = 0 and F −1 (0) = 0, hence proper, but obviously satisfies φ(F) = 0. If X contains a complemented infinite-dimensional subspace with a Schauder basis, it may be arranged in addition that Lip(g) ≤ ε and Lip( f ) ≤ 2 + ε.
Proof. To prove (a), we imitate the construction of Example 3.2 in a more general setting. By a theorem of Banach (see, e.g., [27]), we find an infinite-dimensional closed subspace X 0 ⊆ X with a Schauder basis (e n ) n , e n = 1. If we even find such a space complemented, let P : X → X 0 be a bounded projection. In general, the set B(X 0 ) = X 0 ∩ B(X) is separable, convex, and complete, and so by [31] we may extend the identity map I on B(X 0 ) to a continuous map P : B(X) → B(X 0 ). In both cases, we have P(x) = x for x ∈ B(X 0 ) and P(B(X)) ⊆ B C (X 0 ) for some C ≥ 1. Let c n ∈ X * 0 be the coordinate functions with respect to the basis (e n ) n , and choose µ n > 0 with uniformly on B(X), and since L n (B(X)) and R(B(X)) are bounded subsets of finitedimensional spaces, it follows that g(B(X)) is precompact. Clearly, for x ∈ B(X), and if P is linear, we have also This implies that g(B(X)) ⊆ B ε (X) and, if the subspace X 0 is complemented, then also Lip(g) ≤ ε.
We show now that g(x) = 0 for all x ∈ B(X). In fact, g(x) = 0 implies that L(x) = R(x) ∈ X 0 and so, since (e n ) n is a basis, that µ n c n (P(x)) = (1 − P(x) )µ n for all n. In view of µ n > 0, this means that c n (P(x)) = 1 − P(x) , which shows that c n (P(x)) is actually independent of n. Since P(x) ∈ X 0 , this is only possible if P(x) = 0 which contradicts the equality c n (P(x)) = 1 − P(x) . So, we have shown that g(B(X)) ⊆ B ε (X) \ {0}.
We still have to prove that the equation g(x) = λx has no solution with λ > 0 and x = 1. Assume by contradiction that we find such a solution (λ,x) ∈ (0,∞) × S(X). Since g(x) ∈ X 0 and x = 1, we must have P(x) = x ∈ X 0 , say (3.9) But the relation x = 1 also implies that R(x) = 0, and so the equality g(x) = λx becomes λx + L(x) = 0. Writing this in coordinates with respect to the basis (e n ) n , we obtain, in view of c n (P(x)) = c n (x) = ξ n , that λξ n + µ n ξ n = 0. But from λ + µ n > 0, we conclude that ξ n = 0 for all n, that is, x = 0, contradicting x = 1.
We make some remarks on Theorem 3.3. Although the above construction works in any (infinite-dimensional) Banach space, the completeness of X (at least that of X 0 ) is essential. Moreover, in such spaces uniform limits of finite-dimensional operators must have a precompact range, but it is not clear whether or not they have a relatively compact range. The construction of fixed point free maps in [32] does not have this flaw. Moreover, the maps considered in [32] have even stronger compactness properties, because they send "most" sets (except those of full measure of noncompactness) into relatively compact sets.

Connections with Banach space geometry
The operator g constructed in the proof of Theorem 3.3(a) may be used to show that R φ (X) = 1 in many spaces. To be more specific, we recall some definitions from Banach space geometry. Recall that a space X with (Schauder) basis (e n ) n is said to have a monotone norm (with respect to (e n ) n ) if for all sequences (ξ k ) k and (η k ) k for which the two series on the right-hand side of (4.2) converge.
A basis (e n ) n in X is called unconditional if any rearrangement of (e n ) n is also a basis. Banach spaces with an unconditional basis have some remarkable properties: for example, they are either reflexive, or they contain an isomorphic copy of 1 or c 0 . So, there are many Banach spaces with a Schauder basis but without an unconditional basis. In fact, no space with the so-called Daugavet property has an unconditional basis [20,34]. Moreover, no space with the Daugavet property embeds into a space with an unconditional basis [21]. In particular, C[0,1] and L 1 [0,1] (and all spaces into which they embed) do not possess an unconditional basis.
The following proposition relates spaces with unconditional bases and spaces with monotone norm and seems to be of independent interest. Proof. Assume first that X has an equivalent norm · which is monotone with respect to the basis (e n ) n . Let (η n ) n be such that ∞ k=1 η k e k converges, and assume that |ξ k | ≤ |η k | for all k. Applying (4.1) with ξ k = η k := 0 for k < m ≤ n, we obtain n k=m ξ k e k ≤ n k=m η k e k (m ≤ n), (4.3) and so the Cauchy criterion implies the convergence of ∞ k=1 ξ k e k . Conversely, suppose that the basis (e n ) n is unconditional. Let c n ∈ X * be the corresponding coordinate functionals, and define A n : ∞ × X → X by (4.4) Since the basis (e n ) n is unconditional, by assumption, we have and so the uniform boundedness principle implies that with some finite constant C. This, together with the obvious estimate x ≤ x * , implies that the two norms · and · * are equivalent. Clearly, · * is a norm which satisfies the monotonicity condition (4.1), and so the proof is complete.
Theorem 4.2. Let X be an infinite-dimensional Banach space whose norm is monotone with respect to some basis (e n ) n . Then, the equality holds.
Proof. Consider the map g : with R and L as in (3.5). We already know that g is compact and g(x) = λx for λ > 0 and all x ∈ S(X). Define σ : B(X) → X as in (1.3). Then, σ(x) = 0 on B(X). Indeed, the assumption σ(z) = 0 leads to g(e) = λe, with λ and e defined as in (1.4), a contradiction. So, the map ρ(x) := ν(σ(x)) is a retraction from B(X) onto S(X).
Since g is compact, for any M ⊆ B(X) the set σ(M ∩ B r (X)) is precompact, and so also the set ρ(M ∩ B r (X)). Consequently, Putting by the monotonicity property (4.1) of the norm in X, we conclude that σ(x) ≥ h( x ). Now we distinguish two cases. We assume first that there is a sequence (x n ) n in M \ B r (X) with σ(x n ) → 0 as n → ∞. In view of σ(x) ≥ h( x ) and the definition of h, we obtain then x n → r. Moreover, the definition of σ implies L(x n ) → 0 as n → ∞. Denoting by P k the canonical projection of X onto the linear hull of {e 1 ,...,e k }, we have P k x n → 0, as n → ∞, hence sup n I − P k x n ≥ limsup n→∞ I − P k x n = r (k = 1,2,3,...). (4.11) This implies that γ({x 1 ,x 2 ,x 3 ,...}) ≥ r, and so γ(M) ≥ r ≥ rγ(ρ(M)). Assume now that there is no sequence (x n ) n as above. Then we find a constant c > 0 (possibly depending on r and M) such that Being L a compact operator, it follows that K is contained in a compact set. For x ∈ M \ B r (X), we have and thus In all cases, we conclude that Since r ∈ (0,1) is arbitrary, we see that R γ (X) ≤ 1 as claimed.
The proof of Theorem 4.2 shows that an analogous estimate of the form R φ (X) ≤ C(φ)φ(B(X)) holds for any measure of noncompactness φ on X with the property that for some C(φ) > 0. Some estimates, or even explicit formulas, for the minimal constant C(φ) in some important Banach spaces may be found in [2,Chapter 2].
In view of the above proposition, one might think that it suffices to require in Theorem 4.2 that the basis (e n ) n be unconditional, by passing then, if necessary, to an equivalent norm which is monotone with respect to this basis. Unfortunately, in this case the unit sphere will change, and so the constant R φ (X) will usually change as well. In this connection, the following question arises: given two equivalent norms · and · * on X with corresponding unit spheres S(X) and S * (X), do there exist a constant c > 0 and a homeomorphism ω : S(X) → S * (X) such that φ(ω(M)) = cφ(M) for all M ⊆ S(X)? If the answer is affirmative, then Theorem 4.2 holds true if the basis (e n ) n in X is merely unconditional. We do not know, however, whether or not such a homeomorphism may be found in every space X.
We briefly recall an application of Theorem 4.2 to a long-standing open problem in nonlinear spectral theory which was solved quite recently by Furi [12]. A map f : B(X) → X is called 0-epi [15] if f (x) = 0 on S(X) and, given any compact map g : B(X) → X which vanishes on S(X), one may find a solution x ∈ B(X) of the coincidence equation . More generally, f is called k-epi (k > 0) if this solvability result still holds true for noncompact right-hand sides g satisfying α(g) ≤ k. In this terminology, Schauder's fixed point theorem asserts that the identity operator is 0-epi, and Darbo's fixed point theorem asserts that the identity operator is k-epi for k < 1. It was an open question for some time to find a Banach space X and a map which is 0-epi on B(X), but not k-epi for any positive k. This problem was solved quite recently by Furi [12] by means of an explicit retraction ρ : 1], defined by f (x) := x x, is obviously 0-epi, by Schauder's fixed point theorem. However, it is not k-epi on B(C[0,1]) for any positive k, as may be seen by considering the noncompact right-hand side for sufficiently large n ∈ N. Theorem 4.2 shows that such a construction is possible not only in the space C[0,1], but in any infinite-dimensional space X with monotone norm.

Asymptotically regular maps
Sometimes it is interesting to find maps without fixed points or eigenvalues which have some additional properties. One particularly important class in metric fixed point theory is that of asymptotically regular maps f , that is, those satisfying Proof. Define f as in the proof of Theorem 3.3 (with P(x) = x and C = 2). We claim that, in view of the monotonicity of the norm in X with respect to the basis (e n ) n , the formula (3.10) may be replaced by the simpler formula In fact, for x = ∞ n=1 ξ n e n ∈ B(X), we have and so the monotonicity of the norm implies, in view of 0 ≤ µ n ≤ ε ≤ 1, that . This proves (5.2).
It remains to show that f is asymptotically regular. From (5.2) it follows that g(x) = f (x) − x, and so g( f n (x)) = f n+1 (x) − f n (x). Since g is compact, this implies that the set { f n+1 (x) − f n (x) : n = 1,2,...} ⊆ g(B(X)) is precompact for every x. Now, it suffices to show that every subsequence of ( f n+1 (x) − f n (x)) n contains in turn a subsequence converging to 0. Since we have seen that each subsequence contains a convergent subsequence, we only have to show that the corresponding limit cannot be different from 0. In other words, we must prove that c i ( f n+1 (x)) − c i ( f n (x)) → 0, as n → ∞, where c i (y) denotes the ith coordinate of y as before.
We claim that lim n→∞ f n (x) = 1 (5.5) for every x ∈ B(X). Indeed, one may easily show by induction that For ε ∈ (0,1) we denote by b(ε;n) the set of all indices j ∈ {1, 2,...,n} such that Now, we prove (5.5) by contradiction. If (5.5) is not true, we may find an infinite sequence of numbers (n k ) k (which may depend on ε) such that f nk (x) < 1 − ε for all k. By definition of (5.7), we have Now, we distinguish two cases. Suppose first that the sequence (n k+1 − n k ) k is bounded.
Passing to a subsequence, if necessary, we may then suppose that lim k→∞ n k+1 − n k =: c.
Since the sequence (β(ε,i,n k )) k is bounded, we may also assume, without loss of generality, that the limit β(ε,i) := lim k→∞ β ε,i,n k (5.10) exists. Letting k in (5.8) tend to infinity yields β(ε, By L'Hospital's rule we see that On the other hand, from (5.6) it follows that contradicting the fact that f n (x) ≤ 1 for all n. Suppose now that the sequence (n k+1 − n k ) k is unbounded, and so lim k→∞ n k+1 − n k = ∞. (5.14) Consequently, for some fixed ε > 0, we have then f nk(ε) (x) < 1 − ε (5.15) for an infinite sequence of indices (n k (ε)) k depending on ε. By (5.14) (with ε replaced by ε/3) we find k 0 ∈ N such that n k+1 (ε/3) − n k (ε/3) > 3 for k ≥ k 0 . Taking into account the definition of f , we conclude that Therefore, if we assume that 1 − f nk−1 (x) ≤ ε/3 and 1 − f nk−2 (x) ≤ ε/3, then 1 − f nk (x) ≤ ε. But this contradicts the estimate (5.15), and so we arrived in both cases at a contradiction. This shows that our assumption was false, that is, (5.5) is true. Consequently, combining (5.5) and (5.6), we conclude that lim n→∞ c i f n (x) = 0 (5.17) for every i, and so the proof of the asymptotic regularity of f is complete.

The minimal displacement
Given a normed space X and a map f : B(X) → X, recall that the minimal displacement of f on B(X) is defined by η( f ) := inf x ≤1 x − f (x) . (6.1) Clearly, η( f ) > 0 implies that f has no fixed point, but the converse is true in general only in finite dimensions. For instance, in Kakutani's example (1.9) we have η( f ) = 0. We point out that, by the classical Birkhoff-Kellogg theorem (see, e.g., [10]) for compact maps, the operator g constructed in Theorem 3.3(a) must satisfy inf x ≤1 g(x) = 0. (6.2) From this it follows in turn that the fixed point free operator f from Theorem 3.3(b) satisfies η( f ) = 0. This is not accidental. In fact, in [14] the following remarkable connection between the minimal displacement (6.1) and the α-norm α( f ) of f is given, which may be proved quite easily, even for φ ∈ {α, β,γ}: Theorem 6.1. Let X be a Banach space and suppose that f : B(X)→ B(X) satisfies φ( f )<∞. Then, In particular, φ( f ) ≤ 1 implies η( f ) = 0. x − z + z − y = Lip(ρ) r x − y . (6.10) We already used several times the fact that in each infinite-dimensional normed space X there is a Lipschitz continuous retraction ρ of the unit ball onto its boundary. Using the shortcut k := Lip( f ) and c := Lip(ρ) we have, in particular, and so we get the surprising consequence thatL φ (X) = 1 in every infinite-dimensional normed space, even if we would have replaced φ( f ) by Lip( f ) in the definition (6.5) of L φ (X). Note that the above calculation means in a sense that the estimate (6.3) in Theorem 6.1 becomes "arbitrarily sharp" in each space if η( f ) is sufficiently close to 1, even if we replace φ( f ) by Lip( f ).