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

A universal theorem for stability of ε-isometries of Banach spaces Lixin Cheng ∗,1 , Qingjin Cheng 2 , Kun Tu, Jichao Zhang School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

a r t i c l e

i n f o

Article history: Received 23 July 2014 Accepted 15 April 2015 Available online 5 May 2015 Communicated by G. Schechtman MSC: primary 46B04, 46B20, 47A58 secondary 26E25, 46A20, 46A24 Keywords: ε-isometry Weak stability and strong stability Banach space

a b s t r a c t Let X, Y be two Banach spaces, and f : X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, we show the following sharp weak stability inequality of f : for every x∗ ∈ X ∗ there exists φ ∈ Y ∗ with φ = x∗ ≡ r such that |x∗ , x − φ, f (x) | ≤ 2εr for all x ∈ X. It is not only a sharp quantitative extension of Figiel’s theorem, but it also uniﬁes, generalizes and improves a series of known results about stability of ε-isometries. For example, if the mapping f satisﬁes C(f ) ≡ co[f (X) ∪ −f (X)] = Y , then it is equivalent to the following sharp stability theorem: There is a linear surjective operator T : Y → X of norm one such that T f (x) − x ≤ 2ε, for all x ∈ X; When the ε-isometry f is surjective, it is equivalent to Omladič–Šemrl’s theorem: There is a surjective linear isometry U : X → Y so that f (x) − U x ≤ 2ε, for all x ∈ X. © 2015 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (L. Cheng), [email protected] (Q. Cheng), [email protected] (K. Tu), [email protected] (J. Zhang). 1 Partially supported by NSFC, grant 11371296, and by PhD Programs Foundation of MEC, grant 20130121110032. 2 Partially supported by NSFC, grant 11471271; and by NSFF, grant 2011J01020. http://dx.doi.org/10.1016/j.jfa.2015.04.015 0022-1236/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction Assume that X, Y are Banach spaces. A mapping f : X → Y is said to be an ε-isometry for some ε ≥ 0 provided | f (x) − f (y) − x − y |≤ ε, for all x, y ∈ X.

(1.1)

The mapping f is called an isometry if ε = 0. f is standard if f (0) = 0. In this case we use Yf to denote the subspace spanf (X) of Y . The study of properties of isometries and ε-isometries between Banach spaces has continued for over eighty years since the Mazur–Ulam celebrated theorem ([24], 1932): Every surjective isometry between two Banach spaces is necessarily aﬃne. For general isometries, Figiel showed the remarkable result in 1968 [14]: Every standard isometry from a Banach space to another Banach space admits a linear left-inverse of norm one. Godefroy and Kalton ([16], 2003) resolved a long standing problem about the relation between the existence of isometries and linear isometries. Hyers and Ulam ([19], 1945) ﬁrst studied ε-isometries and proposed a problem, which can be reformulated as follows [25]: Given two Banach spaces X, Y , whether there exists a constant γ > 0 satisfying that for every surjective standard ε-isometry f : X → Y there is a linear surjective isometry U ∈ B(X, Y ) so that f − U is uniformly bounded by γε on X. After 50 years eﬀorts of a number of mathematicians (see, Hyers and Ulam [18–20], Bourgin [5,6,8,7], Gruber [17], Gevirtz [15], Lindenstrauss [21]), a positive answer with the estimate γ = 2 was ﬁnally achieved by Omladič and Šemrl [25] (see, also, [2, Theore 15.2]). They gave an example of a standard surjective ε-isometry f : R → R showing that γ = 2 is optimal. Thus, Omladič–Šemrl’s theorem can be regarded as a sharp quantitative extension of the Mazur–Ulam theorem [24]. The study of properties of non-surjective ε-isometries has been active since 90’s of the last century (see, for instance, [1,3,9–13,25,27–31]). The question, if every standard ε-isometry f : X → Y admits a linear quasi-left inverse, that is, if there exists T ∈ B(Yf , X) so that T f − Id is uniformly bounded on X seems to be very natural. However, Qian [27] showed that for all ε > 0 every separable Banach space Y admitting an uncomplemented subspace X has an unstable standard ε-isometry form X to Y . Therefore, an aﬃrmative answer for the question would imply that Y is, up to an isomorphism, a Hilbert space [22] (see, also, [23]). This disappointment makes us to search for (1) some weaker stability version and (2) some appropriate complementability assumption on some subspaces of Y associated with the mapping. Recently, Cheng, Dong and Zhang gave a weak stability theorem [10, Lemma 2.4], which can be regarded as a quantitative extension of Figiel’s theorem: Suppose that f : X → Y is a standard ε-isometry. Then for every x∗ ∈ X ∗ there exists φ ∈ Y ∗ with φ = x∗ ≡ r so that |x∗ , x − φ, f (x) | ≤ 4εr for all x ∈ X.

(1.2)

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It has played an important role in the study of stability properties of ε-isometries (see, [1,9–12,31]). Making use of it, Cheng and Zhou [11] further presented a stability characterization of ε-isometries. Since Figiel’s theorem says that every standard isometry is stable, without loss of generality, we can always assume an ε-isometry is standard and with ε > 0. This paper is organized as follows. In the second section, we ﬁrst show a sharp version of Cheng– Dong–Zhang’s weak stability theorem (Theorem 2.3), i.e. constant “4” in (1.2) is replaced by “2”. Motivated by Omladič–Šemrl [25], we show the constant “2” in the estimate above is optimal (Theorem 2.4). In the third section, we show that if the ε-isometry f satisﬁes that f (X) contains a sublinear growth net of Y , then Theorem 2.3 is equivalent to the following generalized Omladič–Šemrl’s theorem: There is a surjective linear isometry U : X → Y so that f (x) − U x ≤ 2ε, for all x ∈ X.

(1.3)

We show the constant “2” in the estimate above is optimal in the classical sense; and if C(f ) ≡ co[f (X) ∪ −f (X)] = Y , then it is equivalent to the following sharp stability theorem: There is a linear surjective operator T : Y → X of norm one such that T f (x) − x ≤ 2ε, for all x ∈ X.

(1.4)

The letter X will be a real Banach space and X ∗ its dual. BX and SX , resp., denote the closed unit ball and the unit sphere of X, resp. B(X, Y ) stands for the space of all ∗ bounded operators from X to Y , and ∂ · : X → 2X for the subdiﬀerential mapping of the norm · . For a subspace M ⊂ X, M ⊥ presents the annihilator of M , i.e. M ⊥ = {x∗ ∈ X ∗ : x∗ , x = 0 for all x ∈ M }. If M ⊂ X ∗ , then ⊥ M , the pre-annihilator of M is deﬁned as ⊥ M = {x ∈ X : x∗ , x = 0 for all x∗ ∈ M }. Given a bounded linear operator T : X → Y , T ∗ : Y ∗ → X ∗ stands for its conjugate operator. For a subset A ⊂ X (X ∗ ), A, (w∗ -A) and co(A) presents the closure (the w∗ -closure), and the convex hull of A, respectively. For simplicity, we also use A∗∗ to denote the w∗ -closure of A ⊂ X in X ∗∗ . 2. On sharp inequality of weak stability of ε-isometries In this section, we will show the sharp weak stability version of Cheng–Dong–Zhang’s lemma. Before doing this, we ﬁrst establish the following lemma about ε-isometries. Recall that for a non-empty set Ω, a family U of subsets of Ω is said to be a free ultraﬁlter provided (1) ∅ ∈ / U, and ∩{U ∈ U } = ∅; (2) U, V ∈ U =⇒ U ∩ V ∈ U; (3) U ∈ U and U ⊂ V ⊂ Ω =⇒ V ∈ U; and (4) A ⊂ Ω =⇒ either A ∈ U, or, Ω \ A ∈ U. Let U be a free ultraﬁlter, and K be a Hausdorﬀ space. A mapping g : Ω → K is said to be U-convergent to k ∈ K provided for any neighborhood W of k, we have g −1 (W ) ∈ U. In this case, we write limU g = k. Please note that if K is compact then every mapping g : Ω → K is U-convergent.

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Lemma 2.1. Suppose that f : X → Y is an ε-isometry, and U is a free ultraﬁlter on N. Then Φ(x) = w∗ - lim U

f (nx) , ∀x ∈ X, n

(2.1)

deﬁnes an isometry Φ : X → Y ∗∗ . Proof. Clearly, Φ is well-deﬁned since for every x ∈ X, the bounded sequence ( f (nx) n ) ∗ ∗∗ ∗ is relatively w -compact in Y . Given x, y ∈ X, w -lower semi-continuity of the dual norm · of Y ∗∗ implies Φ(x) − Φ(y) = w∗ - lim( U

≤ lim || U

f (nx) f (ny) − ) n n

f (nx) f (ny) − = x − y. n n

On the other hand, according to the weak stability theorem [10, Lemma 2.4], for any x∗ ∈ ∂x − y, there is φ ∈ Y ∗ with φ = x∗ = 1 such that |φ, f (z) − x∗ , z | ≤ 4ε, ∀z ∈ X. We substitute nx for z in the inequality above, and divide its both sides by n. Then |φ, f (nx)/n − x∗ , x | ≤ 4ε/n. Therefore, w∗ -continuity of φ on Y ∗∗ entails x∗ , x = limφ,

f (nx) = φ, Φ(x) . n

x∗ , y = limφ,

f (ny) = φ, Φ(y) . n

U

Analogously, we obtain

U

Thus, Φ(x) − Φ(y) ≥ φ, Φ(x) − Φ(y) = x∗ , x − y = x − y. So that Φ(x) − Φ(y) = x − y, ∀x, y ∈ X.

2

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Note that if f : X → Y is an ε-isometry, then g = −f (−·) is also an ε-isometry. We can obtain another isometry Ψ : X → Y ∗∗ deﬁned by Ψ(x) = w∗ - lim U

−f (−nx) , for all x ∈ X. n

(2.2)

Lemma 2.2. Let X, Y be Banach spaces, f : X → Y be a standard ε-isometry, and let Φ : X → Y ∗∗ be deﬁned by (2.1). If the norm · of X is Gateaux diﬀerentiable at z ∈ X and with dz = x∗ , then there exists φ ∈ ∂Φ(z) ∩ Y ∗ such that |x∗ , x − φ, f (x) | ≤ 2ε, for all x ∈ X.

(2.3)

Proof. Let x∗ ∈ X ∗ and z ∈ X satisfy dz = x∗ , i.e. lim (x + tz − t) = lim+ (z + tx − z)/t = x∗ , x , for all x ∈ X.

t→∞

t→0

(2.4)

Then x∗ = 1. We can assume z = 1. Given x ∈ X, let un (x) = f (x + nz), and let φn ∈ Y ∗ with φn = 1 such that φn , un (x) = un (x). Then un (x) = φn , f (x + nz) = φn , f (x) + φn , f (x + nz) − f (x) ≤ φn , f (x) + f (x + nz) − f (x) ≤ φn , f (x) + n + ε.

(2.5)

Thus, for any w∗ -cluster point φ of (φn ) we have φ ≤ 1 and lim inf (un (x) − n) ≤ φ, f (x) + ε. n

(2.6)

On the other hand, by deﬁnition of ε-isometry we have lim inf (un (x) − n) ≥ lim inf (x + nz − n) − ε n

n

= lim inf n

z + n−1 x − z − ε = x∗ , x − ε. n−1

Therefore, lim inf (un (x) − n) ≥ x∗ , x − ε. n

(2.7)

This combined with (2.6) entails x∗ , x − φ, f (x) ≤ 2ε.

(2.8)

Next, we show that the functional φ in the inequality above is independent of x. In fact, for any t ≥ 0,

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t + ε ≥ f (tz) ≥ φn , f (tz) = φn , f (x + nz) − φn , f (x + nz) − f (tz) ≥ f (x + nz) − f (x + nz) − f (tz) ≥ (x + nz − ε) − (x + (n − t)z − ε ≥ t − 2(x + ε). Therefore, t + ε ≥ φ, f (tz) ≥ t − 2(x + ε), for all t ≥ 0. Divide the inequality above by t > 0. Then limt→∞ φ, f (tz)/t = 1. Thus, for any w∗ -cluster point z ∗∗ ∈ Y ∗∗ of (f (nz)/n)n∈N (say, Φ(z)), we obtain φ, z ∗∗ = 1. Note z ∗∗ ≤ 1 and φ ≤ 1. We have φ ∈ ∂z ∗∗ and z ∗∗ ∈ ∂φ. In particular, φ ∈ ∂Φ(z) and Φ(z) ∈ ∂φ.

(2.9)

Since z ∗∗ is independent of x, φ is necessarily independent of x. Thus, we have shown x∗ , x − φ, f (x) ≤ 2ε, for all x ∈ X.

(2.10)

Note that, in the proof of the inequality (2.10), for the Gateaux diﬀerentiability point z ∈ X, and for any ﬁxed x ∈ X, the functional φ can be chosen to be any w∗ -cluster point of (φn ) satisfying φn , f (x + nz) = f (x + nz), for all n ∈ N. Since φ is independent of x, by putting x = 0, φ can be any w∗ -cluster point of (φn ) satisfying φn , f (nz) = f (nz), for all n ∈ N.

(2.11)

In the following we show x∗ , x − φ, f (x) ≥ −2ε, for all x ∈ X. Given x ∈ X, let ψn ∈ Y ∗ with ψn = 1 such that ψn , f (x + nz) − f (x) = f (x + nz) − f (x). Then un (x) ≥ ψn , un (x) = ψn , f (x + nz) − f (x) + ψn , f (x)

(2.12)

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= f (x + nz) − f (x) + ψn , f (x) ≥ (nz − ε) + ψn , f (x) = n − ε + ψn , f (x) . Since un (x) − n ≤ (x + nz + ε) − n = (x + nz − nz) + ε =

z + n−1 x − z −→ x∗ , x + ε, n−1

for any w∗ -cluster point ψ of (ψn ) we have x∗ , x − ψ, f (x) ≥ −2ε. Note t + ε ≥ f (tz) ≥ ψn , f (tz) = ψn , f (x + nz) − f (x) − ψn , f (x + nz) − f (tz) + ψn , f (x) ≥ f (x + nz) − f (x) − f (x + nz) − f (tz) − f (x) ≥ (nz − ε) − (x + (n − t)z − ε) − (x + ε) ≥ t − 2x − 3ε. Therefore, t + ε ≥ ψ, f (tz) ≥ t − 2x − 3ε, for all t ≥ 0. Divide the inequality above by t > 0. Then limt→∞ ψ, f (tz)/t = 1. Thus, for any w∗ -cluster point z ∗∗ ∈ Y ∗∗ of (f (nz)/n)n∈N we obtain again ψ, z ∗∗ = 1. Note z ∗∗ ≤ 1 and ψ ≤ 1. We have ψ ∈ ∂z ∗∗ and z ∗∗ ∈ ∂ψ. In particular, ψ ∈ ∂Φ(z), and Φ(z) ∈ ∂ψ.

(2.13)

Since z ∗∗ is independent of x, ψ is necessarily independent of x. Consequently, x∗ , x − ψ, f (x) ≥ −2ε, for all x ∈ X.

(2.14)

Note that, in the proof of the inequality (2.14), for the Gateaux diﬀerentiability point z ∈ X, and for any ﬁxed x ∈ X, the functional ψ can be chosen to be any w∗ -cluster point of (ψn ) satisfying ψn = 1 and φn , f (x + nz) − f (x) = f (x + nz) − f (x), for all n ∈ N.

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Since ψ is independent of x, by putting x = 0, ψ can be any w∗ -cluster point of (ψn ) satisfying ψn , f (nz) = f (nz), for all n ∈ N.

(2.15)

(2.11) and (2.15) together imply that we can take φ = ψ in the inequalities (2.10) and (2.12). Hence, our proof is complete. 2 A Banach space X is said to be a Gateaux diﬀerentiability space provided every continuous convex function deﬁned on a nonempty open convex set D ⊂ X is densely Gateaux diﬀerentiable in D [26]. Please note that every separable Banach space is a Gateaux diﬀerentiability space. A nice characterization for a Banach space X to be a Gateaux diﬀerentiability space is that every nonempty w∗ -compact convex set C ⊂ X ∗ is the w∗ -closed convex hull of its w∗ -exposed points [26, Theorem 6.2]. (Note that x∗ ∈ X ∗ is a w∗ -exposed point of the dual unit ball BX ∗ if and only if x∗ = dx for some Gateaux diﬀerentiability point x ∈ SX .) In particular, a Gateaux diﬀerentiability space X satisﬁes that the closed unit ball BX ∗ of X ∗ is the w∗ -closed convex hull of all Gateaux derivatives {dz : z is a Gateaux diﬀerentiability point of the norm · }. The following theorem is the main result of this paper. Theorem 2.3. Let X, Y be Banach spaces, and f : X → Y be a standard ε-isometry. Then for each x∗ ∈ X ∗ there exists φ ∈ Y ∗ with φ = x∗ ≡ r such that |x∗ , x − φ, f (x) | ≤ 2rε, for all x ∈ X.

(2.16)

Proof. Our proof is divided into three steps. Step I. We ﬁrst show that it is true if X is a Gateaux diﬀerentiability space. Given a Gateaux diﬀerentiability point z ∈ SX , let x∗ = dz. Then by Lemma 2.2 there exists φ ∈ ∂Φ(z) such that (2.16) holds with r = 1. Since X is a Gateaux diﬀerentiability space, for any x∗ ∈ SX ∗ there exist a directed set I, and a net (x∗α )α∈I ⊂ BX ∗ of the form: x∗α =

∗α λα j z j , for each α ∈ I,

j∈Jα

∗ α such that x∗α −w−→ x∗ ; where Jα ⊂ N is a ﬁnite set, λα j ≥ 0 (j ∈ Jα ) satisfy j∈Jα λj = 1; ∗α ∗ α and (z j )j∈Jα are w -exposed points of BX ∗ . Let zj ∈ SX be a Gateaux diﬀerentiability α α ∗ point so that dzjα = z ∗ α j . Then there exists ξj ∈ ∂Φ(zj ) ∩ Y such that α |z ∗ α j , x − ξj , f (x) | ≤ 2ε, for all x ∈ X.

Let φα =

j∈Jα

(2.17)

α λα j ξj . Then we obtain

|x∗α , x − φα , f (x) | ≤ 2ε, for all x ∈ X.

(2.18)

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Consequently, for any w∗ -cluster point φ of (φα ) we have |x∗ , x − φ, f (x) | ≤ 2ε, for all x ∈ X.

(2.19)

Note x∗ = 1 and φ ≤ 1. It is not diﬃcult to observe φ = 1. In fact, since x , z = z = 1, by substituting nz for x in the inequality above, and dividing its both sides by n, we obtain ∗

|1 − φ, f (nz)/n | ≤ 2ε/n. This says φ ≥ limn |φ, f (nz)/n | = 1. Positive homogeneity of (2.19) implies (2.16). Thus, we have shown the theorem in assuming that X is a Gateaux diﬀerentiability space. Step II. In the case that X is a general Banach space, we will show that (2.16) is true for every norm-attaining functional x∗ ∈ X ∗ . Positive homogeneity of (2.16) allows us, without loss of generality, to assume x∗ = 1. Let x0 ∈ SX be such that x∗ , x0 = 1. Let F = {F ⊂ X is a ﬁnite dimensional subspace containing x0 }. Then every element F ∈ F is a Gateaux diﬀerentiability space and x∗ (restricted to F ) is again a norm-attaining functional with x∗ |F = x∗ = x∗ , x0 = 1. Given F ∈ F, by the fact we have just proven in Step 1, there exists φ ∈ SY ∗ such that |x∗ , x − φ, f (x) | ≤ 2ε, for all x ∈ F.

(2.20)

Fix any F ∈ F and let ΦF = {φ ∈ BY ∗ satisfying (2.20)}. Then it is easy to observe that ΦF is a nonempty w∗ -compact convex set. Indeed, nonemptiness of ΦF has been proven by Step I, since F is a Gateaux diﬀerentiability space; convexity and w∗ -compactness of ΦF are trivial by its deﬁnition. Note ΦF ∩ ΦG ⊃ Φspan(F ∪G) for all F, G ∈ F. We obtain that ∩F ∈F ΦF = ∅. Clearly, any φ ∈ ∩F ∈F ΦF is a solution of (2.16) with φ = x∗ = 1. In fact, given φ ∈ ∩F ∈F ΦF , we have φ ≤ 1 and |x∗ , x − φ, f (x) | ≤ 2ε, for all x ∈ X. On the other hand, we replace x by nx0 in the inequality above, divide the two sides by n and notice x∗ , x0 = 1. Then we obtain φ ≥ 1. Therefore, φ = 1. Step III. Finally, we show that the inequality (2.16) holds for every functional x∗ ∈ X ∗ . We can assume again x∗ = 1. By the Bishop–Phelps theorem [4], there is a sequence (x∗n ) ⊂ SX ∗ of norm-attaining functionals such that x∗n → x∗ . By the fact we have just proven in Step II, for each n ∈ N, there exists φn ∈ SY ∗ so that |x∗n , x − φn , f (x) | ≤ 2ε, for all x ∈ X.

(2.21)

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Since x∗n → x∗ , for any w∗ -cluster point φ of (φn ), we have |x∗ , x − φ, f (x) | ≤ 2ε, for all x ∈ X.

(2.22)

Clearly, φ ≤ 1. Conversely, let (zn ) ⊂ SX satisfy 1 = x∗ = limn x∗ , zn . By substituting nzn for x in (2.22) we obtain |x∗ , zn − φ, f (nzn )/n | ≤ 2ε/n, for all n ∈ N. Thus, φ ≥ limφ, f (nzn )/n = 1. n

Consequently, φ = 1.

2

The following result, motivated by [25], says that the constant γ = 2 is optimal. Theorem 2.4. Let X, Y be Banach spaces. If there is a standard ε-isometry g : X → Y for some ε > 0, then for every δ > 0 there is a standard (ε + δ)-isometry f : X → Y such that the following assertion holds: There exist x∗ ∈ SX ∗ and φ ∈ SY ∗ so that |x∗ , x − φ, f (x) | ≤ 2ε + δ, for all x ∈ X

(2.23)

sup |x∗ , x − φ, f (x) | > 2ε − δ.

(2.24)

and

x∈X

Proof. Note that for every n ∈ N gn : X → Y deﬁned by gn (x) = g(nx)/n is a standard ε/n-isometry. Given 0 < δ < ε, let m ∈ N such that h ≡ gm is a standard δ/2-isometry. By Theorem 2.3, for every x∗ ∈ SX ∗ there exists φ ∈ SY ∗ so that |x∗ , x − φ, h(x) | ≤ δ, for all x ∈ X

(2.25)

We ﬁx any point x0 ∈ εSX . Let x∗0 ∈ ∂x0 . By Theorem 2.3 again, there is φ0 ∈ SY ∗ so that |x∗0 , x − φ0 , h(x) | ≤ δ. for all x ∈ X.

(2.26)

Since x∗0 , x0 = ε, ε + δ ≥ φ0 , h(x0 ) ≥ ε − δ. We deﬁne f : X → Y for x ∈ X by ⎧ ⎪ if x = tx0 , t ∈ [0, 1/2]; ⎨ −3th(x0 ), f (x) = (t − 1)h(x0 ), if x = tx0 , t ∈ (1/2, 1]; ⎪ ⎩ h(x), otherwise.

(2.27)

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Then, it is easy to observe that f is a standard (ε + δ)-isometry, and the functionals x∗0 and φ0 satisfy |x∗0 , x − φ0 , f (x) | ≤ 2ε + δ, for all x ∈ X.

(2.28)

Let z = (1/2)x0 in (2.28). Then |x∗0 , z − φ0 , f (z) | > 2ε − δ.

2

(2.29)

Remark 2.5. Figiel’s theorem [14] states that every standard isometry from a Banach space X to another Banach space Y has a linear left-inverse F of norm one. If ε = 0, then Theorem 2.3 deduces for all x∗ ∈ X ∗ , there exists φ ∈ Y ∗ with φ = x∗ such that x∗ , x = φ, f (x) , for all x ∈ X.

(2.30)

The following result says that Theorem 2.3 can be regarded as a sharp quantitative extension of Figiel’s theorem. Theorem 2.6. Suppose that X, Y are two Banach spaces, and f : X → Y is a standard isometry. Let Yf = spanf (X), and the correspondence K : X ∗ → Yf∗ be deﬁned by (2.30), i.e. Kx∗ = φ, where x∗ and φ satisfy (2.30). Then K is a w∗ -to-w∗ continuous linear isometry, which is just the conjugate operator of Figiel’s operator F associated with f . Proof. We ﬁrst claim that the correspondence K : X ∗ → Yf∗ is one-to-one. Given x∗ ∈ X ∗ , assume φ, ψ ∈ Y ∗ such that φ, f (x) = x∗ , x = ψ, f (x) , for all x ∈ X.

(2.31)

Then φ − ψ, f (x) = 0, for all x ∈ X, or, equivalently, φ = ψ on Yf . It is clear the correspondence K deﬁned by (2.30) is homogeneous and additive, i.e. K is linear. Consequently, K is a linear isometry since it is norm-preserving. To show w∗ -to-w∗ continuity of K, let F : Yf → X be Figiel’s operator associated with the isometry f , i.e. the left-inverse of f with F = 1. Then for every x∗ ∈ X ∗ , x∗ , x = x∗ , F (f (x)) = F ∗ (x∗ ), f (x) , for all x ∈ X.

(2.32)

This and (2.30) together imply K(x∗ ), f (x) = F ∗ (x∗ ), f (x) , for all x ∈ X, x∗ ∈ X ∗ . Thus, K = F ∗ , which entails that K is w∗ -to-w∗ continuous. 2

(2.33)

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3. Sharp stability results of a certain class of ε-isometries In this section, we shall see that Theorem 2.3 is useful in the study of stability of ε-isometries. It is not only an extension of Figiel’s theorem, but also a generalized version of the Omladič–Šemrl theorem. A subset N in a metric space (Ω, ) is said to be a sublinear growth net in metric provided for any ﬁxed ω0 ∈ Ω, lim

(ω,ω0 )→∞

(ω, N ) = 0. (ω, ω0 )

(3.1)

For example, let m : R → Z be deﬁned by m(x) = [x] + (signx)[x]P , where [·] denotes the ﬂoor function and [·]P denotes the cardinality of the prime number set P ≡ P (x) = {p ∈ N is a prime number with p ≤ |x|}. Then N ≡ {m(x) : x ∈ R} is a sublinear P growth net of R. In fact, since limx→+∞ [x] ln x = 1, for any ﬁxed x0 ∈ R, [x]P + 1 (x, N ) ≤ → 0, as |x − x0 | → ∞. |x − x0 | |x − x0 | Theorem 3.1. Let X, Y be Banach spaces, and f : X → Y be a standard ε-isometry for some ε ≥ 0. Suppose that f (X) contains a sublinear growth net of Y . Then there is a linear surjective isometry U : X → Y such that f (x) − U x ≤ 2ε, for all x ∈ X.

(3.2)

Proof. Note that if f (X) contains a sublinear growth net of Y , then Yf = Y . We ﬁrst show that this is true if ε = 0, i.e. when f is an isometry. According to Theorem 2.6, the operator K : X ∗ → Yf∗ deﬁned by Kx∗ = φ, where x∗ and φ satisfy (2.30), is just the conjugate of Figiel’s operator, hence, a w∗ -to-w∗ continuous linear isometry. We claim K is surjective. Otherwise, Z ≡ K(X ∗ ) is a w∗ -closed proper subspace of Yf∗ = Y ∗ . In fact, since K : X ∗ → Z is a linear surjective isometry, KBX ∗ = BZ . w∗ -compactness of BX ∗ and w∗ -continuity of K deduce that BZ is w∗ -compact in Y ∗ . Consequently, Z = ∪n∈N nBZ is w∗ -closed in Y ∗ . By separation theorem, there exist ψ ∈ SY ∗ \ K(X ∗ ) and y ∈ SY such that ψ, y = y = 1, and φ, y = 0, for all φ ∈ K(X ∗ ).

(3.3)

Let yn = ny for all n ∈ N. Since f (X) contains a sublinear growth net of Y , for the sequence (yn )n∈N ⊂ Y , there is a sequence (xn )n∈N ⊂ X so that lim n

yn − f (xn ) = 0. n

(3.4)

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Choose any x∗n ∈ ∂xn , and let φn = Kx∗n . Then, 0 = |x∗n , xn − φn , f (xn ) | = |x∗n , xn − φn , f (xn ) − yn | ≥ xn − f (xn ) − yn = f (xn ) − f (xn ) − yn ≥ yn − 2f (xn ) − yn = n(1 − 2

f (xn ) − yn ) −→ ∞. n

This is a contradiction. We have shown that K : X ∗ → Y ∗ is a w∗ -to-w∗ continuous linear surjective isometry. Therefore, its pre-conjugate operator F : Y → X is also a linear surjective isometry satisfying F ◦ f = Id. We are done by letting U = F −1 . ∗ Next, suppose ε > 0. Let : X ∗ → 2Y be deﬁned for x∗ ∈ X ∗ by

x∗ = {φ ∈ Y ∗ : |x∗ − φ ◦ f | is bounded on X};

(3.5)

M = 0 = {φ ∈ Y ∗ : |φ ◦ f | is bounded on X},

(3.6)

and let Q : X ∗ → Y ∗ /M be deﬁned by Qx∗ = x∗ + M.

(3.7)

Then, due to [10, Theorem 4.4], Q is a linear isometry. Since f (X) admits a sublinear growth net of Y , co(f (X)) is dense in Y . Consequently, M = {0}. Note that if φ, ψ ∈ x∗ for some x∗ ∈ X ∗ , then φ − ψ ∈ 0 ⊂ M . Thus, Q : X ∗ → Y ∗ , is actually a single-valued linear isometry. This and Theorem 2.3 together entail |x∗ , x − Qx∗ , f (x) | ≤ 2x∗ ε, for all x ∈ X, x∗ ∈ X ∗ ,

(3.8)

and which further implies that Q : X ∗ → Y ∗ , is a w∗ -to-w∗ continuous linear isometry. Hence, it is a conjugate operator of norm one. Let T : Y → X be a linear operator so that T ∗ = Q. This and (3.7) entail |x∗ , x − x∗ , T f (x) | ≤ 2x∗ ε, for all x ∈ X, x∗ ∈ X ∗ ,

(3.9)

or, equivalently, T f (x) − x ≤ 2ε, for all x ∈ X.

(3.10)

Clearly, T is surjective. In order to show that T is a linear isometry, it suﬃces to prove that Q is surjective. Suppose, to the contrary, that Q(X ∗ ) is a proper subspace of Y ∗ . w∗ -closedness of Q(X ∗ ) implies that there exist ψ ∈ SY ∗ \ (Q(X ∗ )) and y ∈ SY such that ψ, y = y = 1, and φ, y = 0, for all φ ∈ Q(X ∗ ).

(3.11)

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Let again yn = ny for all n ∈ N. Since f (X) contains a sublinear growth net of Y , for the sequence (yn )n∈N ⊂ Y , there is again a sequence (xn )n∈N ⊂ X so that limn yn −fn(xn ) = 0. Choose any x∗n ∈ ∂xn , and let φn = Qx∗n . Then, 2ε ≥ |x∗n , xn − φn , f (xn ) | = |x∗n , xn − φn , f (xn ) − yn | ≥ xn − f (xn ) − yn ≥ (f (xn ) − ε) − f (xn ) − yn ≥ yn − 2f (xn ) − yn − ε = n(1 − 2

f (xn ) − yn ) − ε −→ ∞. n

This contradiction says that Q is surjective. Therefore, we have proven that T : Y → X is a surjective linear isometry. We ﬁnish the proof by letting U = T −1 . 2 The following theorem tells us that the estimate in Theorem 3.1 is sharp. Theorem 3.2. Given a pair of Banach spaces X, Y , if there is a standard ε-isometry g : X → Y satisfying that g(X) contains a sublinear growth net of Y , then there exists a standard ε-isometry f : X → Y with f (X) containing a sublinear growth net of Y such that for every linear surjective isometry U : X → Y we have sup f (x) − U x ≥ 2ε.

(3.12)

x∈X

Proof. Since there is a standard ε-isometry g : X → Y with g(X) containing a sublinear growth net of Y , by Theorem 3.1, there exists a linear isometry U0 : X → Y . We ﬁx any point x0 ∈ εSX , and deﬁne f : X → Y for x ∈ X by ⎧ ⎪ if x = tx0 , t ∈ [0, 1/2]; ⎨ −3tU0 (x0 ), f (x) = (t − 1)U0 (x0 ), if x = tx0 , t ∈ (1/2, 1]; ⎪ ⎩ U (x), otherwise. 0

(3.13)

Then, it is not diﬃcult to see that f is a standard ε-isometry with f (X) containing a sublinear growth net of Y , which satisﬁes that for every linear surjective isometry U :X →Y, sup f (x) − U x ≥ sup f (x) − U0 x ≥ sup f (x) − U0 x = 2ε.

x∈X

x∈X

2

x∈Rx0

Theorem 3.3. Let X, Y be Banach spaces, and f : X → Y be a standard ε-isometry. If C(f ) ≡ co(f (X) ∪ −f (X)) = Y , then there is a linear operator T : Y → X with T = 1 such that T f (x) − x ≤ 2ε, for all x ∈ X.

(3.14)

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213

∗

Proof. Let the mapping : X ∗ → 2Y and the subspace M ⊂ Y ∗ be deﬁned as (3.5) and (3.6). Then C(f ) = Y implies M = {0}. Indeed, given φ ∈ 0, |φ ◦ f | is bounded by some β > 0 on X. This is equivalent to that |φ| is bounded by β on C(f ) = Y . Consequently, φ = 0. Therefore, Q = : X ∗ → Y ∗ is a w∗ -to-w∗ continuous linear isometry, where Q is deﬁned by (3.7). This, incorporating Theorem 2.3, further entails |x∗ , x − Qx∗ , f (x) | ≤ 2x∗ ε, for all x ∈ X, x∗ ∈ X ∗ .

(3.15)

Let T : Y → X be the pre-conjugate operator of Q. Then we obtain T = 1 and x∗ , T f (x) = Qx∗ , f (x) , for all x ∈ X, x∗ ∈ X ∗ .

(3.16)

|x∗ , x − x∗ , T f (x) | ≤ 2x∗ ε, for all x ∈ X, x∗ ∈ X ∗ .

(3.17)

Therefore,

The inequality above is apparently equivalent to (3.14). 2 Remark 3.4. The assumption that C(f ) = Y cannot guarantee the operator T is invertible in Theorem 3.3, even if f is an isometry. For example, let f : X = R → 2∞ = Y be deﬁned by f (x) = (x, ln(1 + x)), if x ≥ 0; = (x, 0), if x < 0. Then f is a standard isometry with C(f ) = 2∞ , so that there is no linear surjective isometry U : X → Y . Acknowledgments The authors are grateful to the referee for his/her encouraging comments and constructive suggestions. They would like to thank Professor Tao Qian and Professor Xiaoping Xue for their supportive discussions on a previous version of the paper, and also to thank people in the FA seminar of Xiamen University for their helpful conversations on this paper. References [1] L. Bao, L. Cheng, Q. Cheng, D. Dai, On universally left-stability of ε-isometry, Acta Math. Sin. (Engl. Ser.) 29 (11) (2013) 2037–2046. [2] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis I, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000. [3] R. Bhatia, P. Šemrl, Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997) 497–504. [4] E. Bishop, R.R. Phelps, A proof that every Banach space is subreﬂexive, Bull. Amer. Math. Soc. 67 (1961) 97–98.

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