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Journal of Functional Analysis ••• (••••) •••–•••

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Journal of Functional Analysis www.elsevier.com/locate/jfa

Stability of almost surjective ε-isometries of Banach spaces Igor A. Vestfrid Nehemya St. 21, 32294 Haifa, Israel

a r t i c l e

i n f o

Article history: Received 15 January 2015 Accepted 7 April 2015 Available online xxxx Communicated by G. Schechtman MSC: 46B04 41A65

a b s t r a c t We show that for every pair of Banach spaces X and Y and for every ε-isometry f : X → Y with sup lim inf dist(ty, f (X))/ y∈SY |t|→∞

|t| < 1/2 there exists an aﬃne surjective isometry V : Y → X such that f (x) − V x ≤ 2ε for all x ∈ X. © 2015 Published by Elsevier Inc.

Keywords: ε-isometry Almost surjectivity Banach spaces Stability

The classical theorem of Mazur and Ulam [7] asserts that a surjective isometry between real normed spaces is aﬃne. Note that it is not valid for complex normed spaces (just consider complex conjugation on C). The hypothesis that an isometry is surjective is essential in general, but can be dropped if the target space is strictly convex, or it can be weakened as it is shown in [4]. As real-world observations have always some minimal error, one may not be able to deduce from measurements whether a given mapping is really isometric or surjective.

E-mail address: [email protected] http://dx.doi.org/10.1016/j.jfa.2015.04.009 0022-1236/© 2015 Published by Elsevier Inc.

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Thus it is natural to ask if a mapping, which only nearly preserves distances and only almost covers the target space, can be well approximated by a surjective (aﬃne) isometry. In this paper we deal with ε-isometries of one Banach space X into another Y which almost cover (in some sense) the target space. Throughout the paper X and Y denote real Banach spaces. Deﬁnition 1. Let ε ≥ 0. A map f : X → Y is called an ε-isometry if f (y) − f (x) − y − x≤ ε for all x, y ∈ X. There is an extensive literature on such mappings starting with the inﬂuential paper [6] of Hyers and Ulam. They proved that every surjective ε-isometry between real Hilbert spaces can be uniformly approximated to within 10ε by an aﬃne surjective isometry. Later this result has been extended to all pairs of real Banach spaces (see [5]), and the constant 10 has been reduced to 2 which is sharp (see [8]). Dilworth [3] showed that the surjectivity condition can be dropped if both above Banach spaces are of the same ﬁnite dimension. However, the example of the map x → (x, 2εx ) from l2n to l2n+1 (which is ε-isometric, but far from any aﬃne map and thus from any isometry) shows that the surjectivity assumption is indispensable in this theorem even for Euclidean spaces. ˘ Semrl and Väisälä [10] showed that every ε-isometry f : X → Y can be uniformly approximated to within 2ε by an aﬃne surjective isometry provided sup {dist (y, f (X))} < ∞.

y∈Y

We show that this result remains true when the above almost surjectivity condition is further relaxed and replaced by sup lim inf dist (ty, f (X)) /|t| < 1/2.

y∈SY |t|→∞

Namely, we give the following theorem. Following [11], given a nonempty Q ⊂ Y and y ∈ SY , we denote (y, Q) = lim inf d(ty, Q)/|t|, |t|→∞

τ (Q) = sup (u, Q). y∈SY

Given a map f : X → Y , we abbreviate (y, f (X)) and τ (f (X)) by (y, f ) and τ (f ). We also abbreviate co(f (X) ∪ −f (X)) by C(f ).

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Theorem 2. Let f : X → Y be an ε-isometry with f (0) = 0. (i) If τ (C(f )) < 1, then there is a surjective norm-one linear operator U : Y → X such that U f (x) − x ≤ 4ε,

x ∈ X.

(1)

(ii) If (y, f ) < 1/2 for every y ∈ SY , then U is injective. We denote its inverse by V := U −1 : X → Y . (iii) If τ (f ) < 1/2, then V is a surjective linear isometry satisfying f (x) − V x ≤ 2ε,

x ∈ X.

Väisälä [11] has posed the following problem: Whether an ε-isometry f : X → Y with τ (f ) = 0 can be approximated by a surjective isometry? Theorem 2(iii) answers this question in aﬃrmative even for τ (f ) < 1/2. Note that in the case when Y is uniformly convex, the weaker condition τ (f ) < 1 implies the existence of such an approximating isometry. It is shown in [12]. We do not know whether the condition (y, f ) < 1/2 for every y ∈ SY is enough to guarantee the existence of an approximating isometry. However, if such an approximating isometry exists, it is necessarily linear and surjective. Proposition 3. Let f : X → Y be an ε-isometry with f (0) = 0 and (y, f ) < 1/2 for every y ∈ SY . Let U : X → Y be an isometry such that U (0) = 0 and U (x) − f (x) = o(x) as x → ∞ uniformly. Then U is a surjective linear isometry and f (x) − U x ≤ 2ε,

x ∈ X.

Proof. Let y ∈ SY . By our assumptions, there are sequences {tn } ⊂ R and {xn } ⊂ X such that |tn | → ∞ and y − U (xn )/tn < 1/2 for all n. Therefore by the theorem of ˘ Figiel, Semrl and Väisälä from [4], U is surjective and linear. Now the result follows by [12, Proposition 2]. 2 In what follows, we shall use some results and notation from [1]. For an ε-isometry f : X → Y with f (0) = 0, we denote by Mε the subspace of Y ∗ consisting of all functionals bounded on C(f ) and by E the annihilator of Mε . Let α ≥ 0. A closed subspace M ⊆ X is said to be α-complemented provided there exist a closed subspace N ⊆ X with M ∩ N = {0} and a projection P : X → M along N such that X = M + N and P ≤ α. It follows from [1, Remark 4.9] and a quick inspection of the proof of [1, Theorem 4.8]: Theorem 4. Let f : X → Y be an ε-isometry with f (0) = 0. Let E be α-complemented in Y and P be a projection P : X → M along N such that X = M + N and P ≤ α. Let

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co(f (X) ∪ −f (X)) ⊂ E + B for some bounded set B ⊂ Y . Then there is a surjective norm-one linear operator U : E → X such that U P f (x) − x ≤ 4ε,

x ∈ X.

Proof of Theorem 2. (i) We prove that Mε = {0}. Then E = Y , and (1) follows by Theorem 4 with P = I (the identity). Choose τ (C(f )) < q < q < 1. Suppose that there is a norm-one ϕ ∈ Mε . Then there is y ∈ SY such that ϕ, y > q . By the deﬁnition of Mε , there is r > 0 such that | ϕ, u| < r for all u ∈ C(f ). Since τ (C(f )) < q , there are sequences {tn } ⊂ R and {un } ⊂ C(f ) such that |tn | → ∞ and y − un /tn < q for all n. Since ϕ = 1, it follows that q > | ϕ, y − un /tn | ≥ | ϕ, y| − | ϕ, un /tn | > q − r/|tn |, which implies q ≥ q – a contradiction. (ii) Let y ∈ SY . We show that U y ≥ 1 − 2(y, f ),

(2)

which implies injectivity of U . Let (y, f ) < q < 1/2. Then there are sequences {tn } ⊂ R and {xn } ⊂ X such that |tn | → ∞ and y − f (xn )/tn < q for all n. Hence f (xn ) > (1 − q )|tn | and then xn ≥ f (xn ) − ε > (1 − q )|tn | − ε. On the other hand, by (1) and U ≤ 1, q |tn | > tn U y − U f (xn ) ≥ xn − |tn |U y − U f (xn ) − xn ≥ xn − |tn |U y − 4ε, which implies xn < (q + U y)|tn | + 4ε. Thus, (1 − q )|tn | − ε < (q + U y)|tn | + 4ε for all n. Thus, U y ≥ 1 − 2q . As q was arbitrary in interval ((y, f ), 1/2), (2) holds. (iii) In this case, U is bijective and U y ≥ 1 − 2τ (f ). Hence its inverse V is bijective and bounded with V ≤ 1/(1 − 2τ (f )). By (1), for every t > 0 tV x − f (tx) ≤ V tx − U f (tx) ≤ 4ε/(1 − 2τ (f )),

x ∈ X.

Hence tV x ≤ f (tx) + 4ε/(1 − 2τ (f )) ≤ tx + 1 +

4 1 − 2τ (f )

ε,

x ∈ X,

which implies V ≤ 1. This along with U ≤ 1 gives that both U and V are isometries. The result follows now by [12, Proposition 2] (or by Proposition 3 above). 2

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Note that conditions in Theorem 2 are rather sharp. The sharpness of the condition in (i) follows from the next two facts: 1. For any mapping f : X → Y with f (0) = 0, we have τ (C(f )) ≤ 1. 2. There exists an ε-isometry with f (0) = 0 such that for any bounded linear operator T:Y → X sup T f (x) − x = ∞

x∈X

(see, for instance, an example presented by Qian in [9]). The sharpness of the conditions in (ii) and (iii) is shown in the following simple example. 2 2 Example 5. Let X = R and Y = l∞ . Deﬁne f : R → l∞ by the formula f (x) = (x, |x|). 2 Then f is a nonlinear isometry and yet, if y = 0 ∈ l∞ then θ1 ty − f ( 12 θ2 ty) ≤ ty/2 for some θ1 , θ2 ∈ {−1, 1} and for all t ∈ R.

Appendix A Concerning Theorem 4, Cheng and Zhou [2] have posed the following problem: Given ε > 0, whether f (X) is always contained in E + B (for some bounded subset B ⊂ Y ) for every ε-isometry? The following lemma gives a negative answer to this question. Lemma 6. For every 1 < p < ∞ and ε > 0, there exists a continuous ε-isometry f : R → lp such that sup dist(f (x), E) = ∞.

x∈R

∞ Proof. Let {ei }∞ i=0 and {ei }i=0 be the canonical bases of lp and lp/(p−1) , respectively. Deﬁne real functions gi by

gi (x) = min

p

pε|x|(p−1) ,i . 2i

Note that p pε |x|(p−1)/p − |y|(p−1)/p |gi (x) − gi (y)| ≤ . 2i p

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Deﬁne f : R → lp by f (x) = xe0 +

∞

gi (x)ei .

i=1

Then |x − y|p ≤ f (x) − f (y)p = |x − y|p +

∞

|gi (x) − gi (y)|p

i=1

p ≤ |x − y|p + pε |x|(p−1)/p − |y|(p−1)/p ≤ |x − y|p + pε|x − y|p−1 < (|x − y| + ε)p . Thus, f is an ε-isometry. For every i > 0 and real x, | ei , f (x)| = gi (x) ≤ i. Hence ei ∈ Mε . It follows that E = span e0 . But for every x, n with |x| ≥ (np 2n /(pε))1/(p−1) , dist(f (x), E) >

n i=1

|gi (x)| =

n i=1

i=

n(n + 1) . 2

Thus, f is a desired mapping. 2 References [1] L. Cheng, Y. Dong, W. Zhang, On stability of nonlinear non-surjective ε-isometries of Banach spaces, J. Funct. Anal. 264 (2013) 713–734. [2] L. Cheng, T. Zhou, On perturbed metric-preserved mappings and their stability characterizations, J. Funct. Anal. 266 (2014) 4995–5015. [3] S.J. Dilworth, Approximate isometries on ﬁnite-dimensional normed spaces, Bull. Lond. Math. Soc. 31 (1999) 471–476. ˘ J. Väisälä, Isometries of normed spaces, Colloq. Math. 92 (2002) 153–154. [4] T. Figiel, P. Semrl, [5] J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983) 633–636. [6] D.H. Hyers, S.M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945) 288–292. [7] S. Mazur, S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, Comp. Rend. Paris 194 (1932) 946–948. ˘ c, P. Semrl, On nonlinear perturbations of isometries, Math. Ann. 303 (1995) 617–628. [8] M. Omladi˘ [9] S. Qian, ε-isometric embeddings, Proc. Amer. Math. Soc. 123 (1995) 1797–1803. ˘ J. Väisälä, Nonsurjective nearisometries of Banach spaces, J. Funct. Anal. 198 (2003) [10] P. Semrl, 268–278. [11] J. Väisälä, A survey of nearisometries, in: J. Heinonen, et al. (Eds.), Papers on Analysis: A Volume Dedicated to Olli Martio on the Occasion of His 60th Birthday, in: Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, Univ. Jyväskylä, Jyväskylä, 2001, pp. 305–315. [12] I.A. Vestfrid, Almost surjective ε-isometries of Banach spaces, Colloq. Math. 100 (2004) 17–22.