ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ ...
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Abstract. This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.
1. Introduction We consider the space of Lipschitz functions on the interval [0, 1] with values in a finite dimensional Hilbert space E. Given a Lipschitz function f, its Lipschitz constant L(f ) is defined to be L(f ) = supx6=y
kf (x) − f (y)k . |x − y|
In this paper we denote by Lip([0, 1], E) the set of all Lipschitz functions on [0, 1] with values in E, equipped with the norm kf k = kf (0)k + L(f ). We study surjective isometries T on Lip([0, 1], E) with the property that there exists a unitary operator UT , depending on T , such that T f (0) = UT (f (0)), ∀ f ∈ Lip([0, 1], E). We show that under some constraints on the surjective isometry T there exists a unitary operator on E, VT and a surjective isometry ψ on [0, 1] such that T has the representation (1)
T f (x) = UT (f (0)) + VT [f (ψ(x)) − f (ψ(0))] ,
Date: June 30, 2011 . 2000 Mathematics Subject Classification. Primary 46B04; Secondary 46E40. Key words and phrases. Spaces of vector valued Lipschitz functions; surjective isometries; isometric embedding of a space of Lipschitz functions into C(Ω); extreme points of the unit ball of the dual of vector valued Lipschitz function spaces. The research of Bentuo Zheng is partially surported by NSF DMS-1068838 and NSF DMS-0800061. 1
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
for every f ∈ Lip([0, 1], E). We note that there are only two surjective isometries on [0, 1], the identity and the reflection. Furthermore,operators of the form described in (1), acting on Lip(X, E) with X a compact metric space, E an arbitrary Banach space and UT and VT isometries on E, define a group of isometries under composition. This is a subgroup of the isometry group of Lip(X, E). The isometries supported by a Banach space reflect important geometric properties of the space, as a result a characterization of its isometry group is of great interest. Such study is traced back to the pioneering work of Banach and Stone, who obtained the first characterization for the surjective isometries between spaces of scalar valued continuous functions. This well-known result is referred to as the Banach-Stone theorem, see [2, 14]. Researchers have derived extensions of the Banach-Stone theorem for several different settings. We refer the reader to [8] and [9] for a survey of the topic. Cambern and Pathak in [4] and [5], have considered isometries on spaces of scalar valued differentiable functions and determined a representation for the surjective isometries supported by such spaces. Rao and Roy in [13] described the isometries of Lipschitz spaces of scalar valued functions with a variety of different norms. Vargas and Vallecillos in [11] have also considered isometries of spaces of vector valued Lipschitz functions from a compact metric space with values on a strictly convex Banach space, equipped with the norm kf k = max{L(f ), kf k∞ }, see also [10]. In their work, Vargas and Vallecillos require the isometries to satisfy certain natural conditions, similar constraints are also required in our work. The authors in [3] gave a similar representation to the one in [11] for surjective isometries of spaces of vector valued Lipschitz functions from a compact metric space with values on quasi sub-reflexive Banach space with trivial centralizer. In this paper, we extend the results of Rao and Roy in [13] to the vector valued Lipschitz function spaces. Representations for surjective isometries on certain Banach spaces are often derived from the action of their adjoints on the set of the extreme points of the unit ball of the corresponding dual space. A complete description of the extreme points of the dual ball of a Banach space is often very hard to obtain. However a characterization of the extreme points of the dual unit ball of closed subspaces of continuous
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
3
functions was found by Arens and Kelley. This result is a crucial tool in the characterization of the surjective isometries on function spaces. The result of Arens and Kelley is stated in the following theorem.
Theorem 1.1. (cf. [7] pg.441, see also [1]) If X is a closed subspace of C(Ω), which consists of all real or complex valued continuous functions defined on the compact Hausdorff space Ω, and for each ω ∈ Ω, δω ∈ X ∗ is given by δω (f ) = f (ω), f ∈ X , then every extreme point of the unit ball in X ∗ is of the form λδω , with |λ| = 1. If X = C(Ω) the converse also holds, i.e. every element of the form λδω , with |λ| = 1 and ω ∈ Ω, is an extreme point of the unit ball in X ∗ .
The standard approach to the study of the isometries of a function space relies on an isometric embedding of the function space into C(Ω), with Ω a compact Hausdorff space. We adapt this technique in order to embed Lip([0, 1], E) into C(Ω), for a convenient compact and Hausdorff space Ω. Then we proceed with the analysis of the extreme points of Lip([0, 1], E) now identified with a closed subspace of C(Ω). This study leads to natural constraints on a surjective isometry T which allows a representation for T , as an affine weighted composition operators.
2. An isometric embedding of Lip([0, 1], E) into C(Ω) In this section we construct an isometric embedding of Lip([0, 1], E) (with E a finite dimensional complex Banach space) into a space of scalar valued continuous functions on a compact topological space. We denote ˜ = [0, 1] × [0, 1] \ {(x, y) : y ≥ x} by E1∗ the unit ball of the dual space of E with the norm topology. We set X ˜ the Stone-Cech ˇ ˜ Given a function f ∈ Lip([0, 1], E) let f˜ : X ˜ →E and denote by β(X) compactification of X. be defined by f˜(x, y) =
f (x)−f (y) . x−y
˜ Then f β denotes the unique continuous extension of f˜ to β(X).
Lemma 2.1. Let E be a finite dimensional Banach space. Then Lip([0, 1], E) is isometric to a closed ˜ subspace of C(E1∗ × E1∗ × β(X)).
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
˜ as follows: Proof. We define ı : Lip([0, 1], E) → C(E1∗ × E1∗ × β(X)) ˜ ı(f ) = F with F (v ∗ , w∗ , η) = v ∗ (f (0)) + w∗ (f β (η)), ∀(v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X). We show that ı is an isometric isomorphism. The linearity follows from the uniqueness of the extension of ˜ We now prove that ı is an isometry, i.e. for every f ∈ Lip([0, 1], E) f˜ to β(X). ∗ ∗ β kf k = kf (0)k + L(f ) = max(v∗ ,w∗ ,η)∈E ∗ ×E ∗ ×β(X) ˜ |v (f (0)) + w (f (η))|. 1
1
If f (0) = 0, let v ∗ be any functional in E1∗ . If f (0) 6= 0, let v ∗ be a norm 1 functional on E such that ˜ such that v ∗ (f (0)) = kf (0)k. Let {(xα , yα )} be a net in X
L(f ) = lim α
kf (xα ) − f (yα )k . xα − yα
˜ Let η = limα {(xα , yα )}. For each α, we define wα∗ ∈ E ∗ The net {(xα , yα )} has a converging subnet in β(X). 1 to be a norm 1 functional on E such that wα∗
f (xα ) − f (yα ) xα − yα
=
kf (xα ) − f (yα )k . xα − yα
The net of functionals {wα∗ } has a subnet that converges to w0∗ in E1∗ . Then kf k = v ∗ (f (0)) + w0∗ (f β (η)) = kı(f )k∞ . The Closed Graph Theorem asserts that the range of ı is a closed subspace of C(Ω). This concludes the proof.
We denote by R the range of ı, and we observe that ı induces the following isometry between the corresponding dual spaces: ı∗ : R∗ F∗ defined by ı∗ (F ∗ )(g) = F ∗ (ı(g)).
→ Lip([0, 1], E)∗ ı∗ (F ∗ )
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
5
The isometry ı∗ maps the extreme points of R∗1 , the unit ball in R∗ onto the extreme points of the unit ball in the dual space of Lip([0, 1], E). We denote the set of all extreme points of a convex set A by ext(A).
3. Existence of extreme points of Lip([0, 1], E)∗1 In this section we study the extreme points of the unit ball of the dual of Lip([0, 1], E). We identify Lip([0, 1], E) with its isometric image under ı, i.e. R = ı(Lip([0, 1], E)). Theorem 1.1 asserts that ˜ ∗1 ) = {δω : ω ∈ E1∗ × E1∗ × β(X), ˜ with δω (g) = g(ω), for all g}, ext(C(E1∗ × E1∗ × β(X)) ˜ hence the extreme points of R∗1 are functionals of the form δ(v∗ ,w∗ ,η) , with (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X), ext(R∗1 ) ⊆ {δ(v∗ ,w∗ ,η) : δ(v∗ ,w∗ ,η) (F ) = v ∗ (f (0)) + w∗ (f β (η)), ∀f ∈ Lip([0, 1], E)}, (F = ı(f )). Our first lemma shows that the interdependence between the extreme points of Lip([0, 1], E)∗1 and the points ˜ in the growth of β(X).
˜ \ X}. ˜ Lemma 3.1. ext(R∗1 ) ⊆ {δ(v∗ ,w∗ ,η) : (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X) ˜ v ∗ and w∗ ∈ E ∗ , we show that δ(v∗ ,w∗ ,(x,y)) ∈ Proof. Let (x, y) ∈ X, / ext(R∗1 ). We recall that δ(v∗ ,w∗ ,(x,y)) (F ) = 1 (y) . We define δ1 and δ2 in R∗ as follows: v ∗ (f (0)) + w∗ f (x)−f x−y ∗
δ1 (F ) = v (f (0)) + 2w
∗
f (x) − f ( x+y 2 ) x−y
! = δ(v∗ ,w∗ ,(x, x+y )) (F ) 2
and ∗
δ2 (F ) = v (f (0)) + 2w We note that δ(v∗ ,w∗ ,(x,y)) = Given f (t) = (t − x) (t −
∗
f ( x+y 2 ) − f (y) x−y
! = δ(v∗ ,w∗ ,( x+y ,y)) (F ). 2
δ1 +δ2 2 .
x+y 2 ) w,
for every t ∈ [0, 1] and w∗ (w) = 1. We have δ1 (F ) =
δ2 (F ) = δ1 (F ) − (x − y). Therefore δ1 6= δ2 .
x2 +xy ∗ v (w) 2
and
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Now we show that kδ1 k ≤ 1. Similar reasoning applies to δ2 . For every f ∈ Lip([0, 1], E), |δ1 (F )| ≤ |v (f (0))| + 2 w∗
f (x) − f ( x+y 2 ) x−y
∗
! ≤ kf (0)k + L(f ) = kf k = kF k∞ .
These considerations show that δ(v∗ ,w∗ ,(x,y)) is not an extreme point of R∗1 .
Lemma 3.2. Let E be a finite dimensional Hilbert space and let δ(v0∗ ,w0∗ ,η0 ) be an extreme point of R∗1 , with ˜ \ X. ˜ Then for every v ∗ and w∗ ∈ E ∗ , δ(v∗ ,w∗ ,η ) is an extreme point of R∗ . v0∗ , w0∗ ∈ E1∗ and η0 ∈ β(X) 1 1 0 Proof. We fix v ∗ and w∗ ∈ E1∗ . Let {u1 , u2 , . . . un } and {u01 , u02 , . . . u0n } be orthonormal bases for E such that u1 = v0 and u01 = v. We define the unitary operator U =
Pn
i=1
ui ⊗ u0i , with ui ⊗ u0i (u) = hu, u0i iui , for every
u ∈ E. We notice that U v = v0 . We first show that δ(v∗ ,w0∗ ,η0 ) is an extreme point of R∗1 . Suppose otherwise, then δ(v∗ ,w0∗ ,η0 ) =
δ1 + δ2 , δ1 , δ2 ∈ R∗1 and δ1 6= δ2 . 2
For j = 1, and 2 we set δj1 : R F
→
C
→ δj (ı(f1 )),
with f1 (x) = f (x) + U ∗ f (0) − f (0), (U ∗ denotes the adjoint of U ). We show that δ11 and δ21 are distinct functionals in R∗1 . This leads to a contradiction, since δ(v0∗ ,w0∗ ,η) =
δ11 +δ21 2 .
We observe that there exists F ∈ R such that δ1 (F ) 6= δ2 (F ). We set g = f + U f (0) − f (0), hence δ11 (ı(g)) = δ1 (F ) 6= δ2 (F ) = δ21 (ı(g)), proving that δ11 6= δ21 . It is straightforward to check that kδj1 k ≤ 1 (j = 1, 2), |δj1 (F )| = δj (ı(f ))| ≤ kı(f )k∞ = kF k∞ = kf k. This contradiction shows that for every v ∗ ∈ E1∗ , δ(v∗ ,w0∗ ,η0 ) is an extreme point of R∗1 . We assume that δ(v∗ ,w0∗ ,η0 ) ∈ ext(R∗1 ) and prove that δ(˜v∗ ,w∗ ,η0 ) is also in ext(R∗1 ), for some v˜∗ ∈ E1∗ conveniently defined. We consider the following two orthonormal bases for E, {u1 , u2 , . . . un } and {u01 , u02 , . . . u0n }
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
such that u1 = w0 and u01 = w and the unitary operator V =
Pn
i=1
7
u0i ⊗ ui . We have that w∗ (u) = w0∗ (V ∗ u)
and we define v˜∗ ∈ E1∗ as follows: v˜∗ (u) = v0∗ (V ∗ u), where V ∗ stands for the adjoint of V . We assume that δ(˜v∗ ,w1∗ ,η0 ) is not an extreme point of R∗1 . Thus there exist distinct functionals in R∗1 , δ1 and δ2 such that
δ(˜v∗ ,w∗ ,η0 ) =
δ1 + δ2 . 2
We define δ˜j (F ) = δj (ı(V f )) (j = 1, 2) and V f (x) = V (f (x)), for every x ∈ [0, 1]. It is straightforward to check that δ(v0∗ ,w0∗ ,η0 ) =
δ˜1 +δ˜2 2 .
This also leads to a contradiction, since δ˜1 6= δ˜2 and kδ˜i k ≤ 1 (i = 1, 2). This
contradiction shows that δ(˜v∗ ,w∗ ,η0 ) is an extreme point of R∗1 , and the first part of the proof implies that δ(v∗ ,w∗ ,η0 ) must also be an extreme point.
Remark 3.1. Since the set of extreme points of R∗1 is nonempty, then there exists a triplet (v0∗ , w0∗ , η0 ) for which δ(v0∗ ,w0∗ ,η0 ) is an extreme point, and an application of Lemma 3.2 implies that ext(R∗1 ) ⊇ {δ(v∗ ,w∗ ,η0 ) : v ∗ , w∗ ∈ E1∗ }.
4. The trace of an extreme point in [0, 1] ˜ Throughout this section we fix v0∗ and w0∗ in E1∗ . We define the following subset of β X
˜ : δ(v∗ ,w∗ ,η) ∈ ext(R∗1 )}. A = {η ∈ β X 0 0 ˜ \X ˜ and an arbitrary net {(xα , yα )} ∈ X ˜ converging to η, it follows from Lemma 3.1 that Given η ∈ β X {xα } and {yα } must converge to the same point x ∈ [0, 1]. We call this point x the trace of δ(v0∗ ,w0∗ ,η) in [0, 1], or the trace of η in [0, 1]. We now define τ : A → [0, 1] given by τ (η) = x, where x is the unique point in [0, 1] determined by η. In this section we investigate properties of τ to be used in forthcoming results. ˜ that yield identical functionals δ(v∗ ,w∗ ,η) , We first characterize those triplets (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X) when restricted to R.
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Given a scalar valued Lipschitz function λ on [0, 1] we denote by λβ the unique continuous extension of λ ˜ We now start with a preliminary result. to β(X).
Lemma 4.1. If η0 and η1 are two distinct points in A such that τ (η0 ) 6= τ (η1 ), then there exists λ ∈ Lip([0, 1], R) such that λβ (η0 ) 6= λβ (η1 ).
Proof. We set x = τ (η0 ) and y = τ (η1 ). We assume that 0 ≤ x < y ≤ 1. We define t for 0 ≤ t ≤ x+y 2 λ(t) = x+y for x+y ≤ t ≤ 1. 2 2 It is easy to see that λβ (η0 ) = 1 and λβ (η1 ) = 0.
˜ with associated Proposition 4.1. Let (v0∗ , w0∗ , η0 ) and (v1∗ , w1∗ , η1 ) be two distinct triplets in E1∗ × E1∗ × β(X) evaluation functionals δ(v0∗ ,w0∗ ,η0 ) and δ(v1∗ ,w1∗ ,η1 ) respectively. Then δ(v0∗ ,w0∗ ,η0 ) (F ) = δ(v1∗ ,w1∗ ,η1 ) (F ), for all F ∈ R if and only if v0∗ = v1∗ , w0∗ = w1∗ and λβ (η0 ) = λβ (η1 ), for every λ ∈ Lip([0, 1], R).
Proof. If δ(v0∗ ,w0∗ ,η0 ) (F ) = δ(v1∗ ,w1∗ ,η1 ) (F ), for all F ∈ R then (2)
v0∗ [f (0)] + w0∗ [f β (η0 )] = v1∗ [f (0)] + w1∗ [f β (η1 )].
We get v0∗ = v1∗ and v0 = v1 by evaluating (2) at constant functions. Therefore (2) reduces to (3)
w0∗ [f β (η0 )] = w1∗ [f β (η1 )], ∀ f ∈ Lip([0, 1], E).
Equation (3) applied to f (x) = x · v, with v ∈ E, implies w0 = w1 . We apply (3) to functions of the form f = λ · w0 , with λ ∈ Lip([0, 1], R) to conclude that λβ (η0 ) = λβ (η1 ), for every λ ∈ Lip([0, 1], R). Since E is finite dimensional, let {vi } be an orthonormal basis for E, then f ∈ Lip([0, 1], E) has the (unique) representation f (x) =
n X i=1
λi (x) vi , with λi ∈ Lip([0, 1], R).
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
9
Therefore f β (η0 ) = f β (η1 ), ∀ f ∈ Lip([0, 1], E) and the reverse implication follows easily. This completes the proof.
Remark 4.1. It follows easily from (3) that τ (η0 ) = τ (η1 ). In fact, if we assume that τ (η0 ) 6= τ (η1 ) Lemma 4.1 implies the existence of λ such that λβ (η0 ) = 1 and λβ (η1 ) = 0. We set f = λv, with v ∈ E. Thus (3) implies that w0∗ (v) = 0, for every v. This contradiction shows that τ (η0 ) = τ (η1 ).
Lemma 4.2. τ (A) is a dense subset of [0, 1].
Proof. Suppose τ (A) is not dense then there exist y ∈ (0, 1) and an open interval I = (y − , y + ) ⊂ (0, 1) such that I
T
τ (A) = ∅. We consider the evaluation functional δ(v0∗ ,w0∗ ,(y+ 4 ,y− 4 )) . This functional is in the
weak*-closure of the convex hull of ext(R∗1 ). We define the following scalar valued Lipschitz function:
α(x) =
0
if 0 ≤ x ≤ y −
x − (y − 2 )
if y −
2
2
≤x≤y+
2
otherwise.
Now we set f (x) = α(x)w0 , with w0∗ (w0 ) = 1. We denote by F the image of f under ı. Therefore |δ(v0∗ ,w0∗ ,(y+ 4 ,y− 4 )) (F )| = |v0∗ (f (0)) + w0∗ (
α(y+ 4 )−α(y− 4 ) /2
w)| = 1.
On the other hand a convex combination of extreme points is of the form
X
λi,j,k δ(vi∗ ,wj∗ ,ηk ) ,
i=1...n1 , j=1...n2 , k=1...n3
with δ(vi∗ ,wj∗ ,ηk ) ∈ ext(R∗1 ) and λi,j,k scalars. Any such convex combination applied to F yields zero. This would contradict the Krein-Milman Theorem and thus τ (A) is dense in [0, 1].
˜ by cl(A). In the next result we denote the closure of A in β(X)
Lemma 4.3. The map τ is continuous and it has a continuous extension to cl(A).
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Proof. Let {ηα }α∈D be a net in A converging to η ∈ A. For each α ∈ D, let τ (ηα ) = xα and τ (η) = x. We show that xα converges to x. We suppose otherwise. Then there exist > 0 and a subnet of {xα }α∈D1 ( with D1 a directed subset of D and cofinal in D) such that, for all α ∈ D1 , xα is in the complement of T (x − , x + ) [0, 1]. We consider the following net of extreme points of R∗1 , {δ(v0∗ ,w0∗ ,ηα ) }α∈D1 . This net converges to δ(v0∗ ,w0∗ ,η) in the weak*-topology, hence we have
lim w0∗ [f β (ηα )] = w0∗ [f β (η)], ∀f ∈ Lip([0, 1], E). α
This equation leads to a contradiction when applied to the function f = λw0 , with λ a real valued Lipschitz T function defined on [0, 1] such that λ restricted to (x − , x + ) [0, 1] is linear with slope 1 and is constant T on the complement of (x − , x + ) [0, 1]. This contradiction proves that τ is continuous. We now show that τ can be continuously extended to cl(A). Let η0 ∈ cl(A) \ A, then there exists a net of points in A converging to η0 , i.e. limα ηα = η0 . As before, we set τ (ηα ) = xα . The net of extreme points {δ(v0∗ ,w0∗ ,ηα ) } converges to δ(v0∗ ,w0∗ ,η0 ) in the weak*-topology. Equivalently we write
lim w0∗ [f β (ηα )] = w0∗ [f β (η0 )], ∀f ∈ Lip([0, 1], E). α
In particular, for f = λ · w0 with λ ∈ Lip([0, 1], R), we conclude that
(4)
lim λβ (ηα ) = λβ (η0 ), ∀λ ∈ Lip([0, 1], R). α
˜ {(xα , y α )}β with limit ηα , i.e. limβ (xα , y α ) = ηα . Lemma 3.1 For each α, there exists a net of points in X, β β β β α asserts that limβ xα β = limβ yβ = xα ∈ [0, 1]. A similar argument used in the first part of this proof also
shows that {xα } converges to some point in [0, 1], say x0 . ˜ then it is the limit of a net in X, ˜ say limγ (cγ , dγ ) = η0 . Without loss of generality we Since η0 ∈ β(X), assume that limγ cγ = a, limγ dγ = b with a 6= b. We first consider a < b, and x0 < a. Equation in (4) leads
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
to an absurd when applied to the function x λ(x) = x0 +a 2
for
0≤x≤
for
x0 +a 2
If x0 = a < b we consider the following function x λ(x) = −x + a + b
11
x0 +a 2
≤ x ≤ 1.
for
0≤x≤
for
a+b 2
a+b 2
≤ x ≤ 1.
Equation (4) also leads to a contradiction. The remaining cases are eliminated similarly, implying that a = b. Therefore τ can be extended to η0 as follows: τ (η0 ) = a. By constructing a Lipschitz function which is a constant on a small neighborhood of x0 and has nontrivial derivative at a, we readily see that a = x0 . The continuity of τ then follows easily.
Remark 4.2. It is important to observe that τ, now extended to cl(A), is surjective. Given x0 ∈ [0, 1] \ τ (A), there exists a net in τ (A) converging to x0 , hence there exists a convergent net {ηα } in A such that τ (limα ηα ) = x0 .
5. A representation for T : The homeomorphism ϕ A surjective isometry T : Lip([0, 1], E) → Lip([0, 1], E) induces an isometric bijection between the extreme points of R∗1 , i.e. T˜∗ (ext(R∗1 )) = ext(R∗1 ). This translates as follows. For every (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × A there exists (˜ v∗ , w ˜ ∗ , η˜) ∈ E1∗ × E1∗ × A such that
(5)
v ∗ (T f (0)) + w∗ ((T f )β (η)) = v˜∗ (f (0)) + w ˜ ∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1, ], E).
We now investigate whether the statement in (5) can be extended to points in E1∗ × E1∗ × cl(A). Given η0 ∈ cl(A), we consider a net in A, {ηα }α such that limα ηα = η0 . Therefore we have v ∗ (T f (0)) + w∗ ((T f )β (ηα )) = v˜α∗ (f (0)) + w ˜α∗ (f β (˜ ηα )), ∀ f ∈ Lip([0, 1, ], E),
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
since T ∗ (δ(v∗ ,w∗ ,ηα ) = δ(˜vα∗ ,w˜α∗ ,˜ηα ) ). We assume that the net of functionals δ(˜vα∗ ,w˜α∗ ,˜ηα ) converges to δ(˜v0∗ ,w˜0∗ ,˜η0 ) , by selecting a convergent subnet. Hence we conclude that lim {v ∗ (T f (0)) + w∗ ((T f )β (ηα ))} = v˜0∗ (f (0)) + w ˜0∗ (f β (˜ η0 )), ∀ f ∈ Lip([0, 1, ], E). α
If another subnet of δ(˜vα∗ ,w˜α∗ ,˜ηα ) converges to a different functional, δ(˜v1∗ ,w˜1∗ ,˜η1 ) with (˜ v1∗ , w ˜1∗ , η˜1 ) ∈ E1∗ × E1∗ × cl(A), then v˜0∗ (f (0)) + w ˜0∗ (f β (˜ η0 )) = v˜1∗ (f (0)) + w ˜1∗ (f β (˜ η1 )), ∀ f ∈ Lip([0, 1, ], E). Proposition 4.1 implies that v0∗ = v1∗ , w0∗ = w1∗ , and for every λ ∈ Lip([0, 1], R), λβ (˜ η0 ) = λβ (˜ η1 ). We then observe that T˜∗ maps extreme points of R∗1 onto extreme points of R∗1 and also point evaluation functionals δ(v∗ ,w∗ ,η) with (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × cl(A) \ A onto point evaluation functionals of the same form. These considerations lead to the following equivalence relation defined on cl(A): η0 ≈ η1 iff δ(v∗ ,w∗ ,η0 ) |R ≡ δ(v∗ ,w∗ ,η1 ) |R , ∀v ∗ and w∗ ∈ E1∗ . We now denote by A˜ the quotient space of cl(A) relative to ≈, i.e. A˜ = cl(A)/ ≈. It follows from Proposition 4.1 that η0 ≈ η1 iff λβ (η0 ) = λβ (η1 ), ∀λ ∈ Lip([0, 1], R). We represent elements in A˜ by [[η]], with η ∈ cl(A). We now consider the mapping ϕ : E1∗ × E1∗ × A˜ → ˜ given by ϕ(v ∗ , w∗ , [[η]]) = (˜ E1∗ × E1∗ × A, v∗ , w ˜ ∗ , [[˜ η ]]) iff v ∗ [T f (0)] + w∗ [(T f )β (η)] = v˜∗ [f (0)] + w ˜ ∗ [f β (˜ η )], for all f ∈ Lip([0, 1], E). Proposition 4.1 also implies that ϕ is well defined. Furthermore, ϕ is a bijection since T is a surjective isometry.
Proposition 5.1. ϕ is a homeomorphism.
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
13
˜ denoted Proof. We start with a net {(vα∗ , wα∗ , [[ηα ]])}α in E1∗ × E1∗ × A˜ converging to a point in E1∗ × E1∗ × A, by (v ∗ , w∗ , [[η]]). The continuity of ϕ then follows provided we show that {ϕ(vα∗ , wα∗ , [[ηα ]])}α converges to ˜ has a convergent subnet. Without loss of generality ϕ(v ∗ , w∗ , [[η]]). The net {(˜ vα∗ , w ˜α∗ , η˜α )} in E1∗ ×E1∗ ×β(X) ˜ we assume that {(˜ vα∗ , w ˜α∗ , η˜α )} converges to (˜ v0∗ , w ˜0∗ , η˜0 ) in E1∗ × E1∗ × β(X). Since ϕ(vα∗ , wα∗ , ηα ) = (˜ vα∗ , w ˜α∗ , η˜α ) we have
(6)
vα∗ [T f (0)] + wα∗ [(T f )β (ηα )] = v˜α∗ [f (0)] + w ˜α∗ [f β (˜ ηα )], ∀ f ∈ Lip([0, 1], E).
Consequently we have lim v˜α∗ [f (0)] + w ˜α∗ [f β (˜ ηα )] = v˜0∗ [f (0)] + w ˜0∗ [f β (˜ η0 )], α
for every f ∈ Lip([0, 1], E), since |˜ vα∗ [f (0)] + w ˜α∗ [f β (˜ ηα )] − v˜0∗ [f (0)] − w ˜0∗ [f β (˜ η0 )]| ≤ kf (0)kk˜ vα∗ − v˜0∗ k + kf β (˜ ηα ) − f β (˜ η0 )k + kw ˜α∗ − w ˜0∗ kkf β (˜ η0 )k.
On the other hand, we also have lim vα∗ [T f (0)] + wα∗ [(T f )β (ηα )] = v ∗ [T f (0)] + w∗ [(T f )β (η)], α
hence (6) implies that v˜0∗ [f (0)] + w ˜0∗ [f β (˜ η0 )] = v ∗ [T f (0)] + w∗ [(T f )β (η)], ∀ f ∈ Lip([0, 1], E). Consequently ϕ(v ∗ , w∗ , [[η]]) = (˜ v0∗ , w ˜0∗ , [[˜ η0 ]]) implying the continuity of ϕ. The continuity of ϕ−1 is a consequence of the surjectivity of the underlying isometry T .
v∗ , w ˜ ∗ , η˜) ∈ We recall that given w∗ ∈ E1∗ and η ∈ A (respectively cl(A) \ A), for every v ∗ ∈ E1∗ there exists (˜ E1∗ × E1∗ × A (respectively E1∗ × E1∗ × cl(A) \ A) such that
(7)
v ∗ (T f (0)) + w∗ ((T f )β (η)) = v˜∗ (f (0)) + w ˜ ∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1, ], E).
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Since T is a surjective isometry satisfying the following property:
(∗) There exists a unitary operator UT s.t. T (f )(0) = UT (f (0)), ∀ f ∈ Lip([0, 1], E), we have that the correspondence assigning v˜∗ to v ∗ is bijective. ˜ ϕi : E ∗ × E ∗ × A → E ∗ , ϕi (v ∗ , w∗ , [[η]]) = The map ϕ defines three natural continuous maps on E1∗ × E1∗ × A, 1 1 1 πi ϕ(v ∗ , w∗ , [[η]]), with i = 1 or 2, and ϕ3 : E1∗ × E1∗ × A˜ → A˜ given by ϕ3 (v ∗ , w∗ , [[η]]) = π3 ϕ1 (v ∗ , w∗ , [[η]]), with πi (i = 1, 2, or 3) denoting the projection onto component i. Equation (7) with f ≡ u, a constant function, yields v ∗ [UT u] = v˜∗ (u). The next lemma follows easily from this observation.
Lemma 5.1. Let E be a finite dimensional Hilbert space, w0∗ ∈ E1∗ and η0 ∈ A. If T is a surjective isometry on Lip([0, 1], E) satisfying property (*), then for every v ∗ , w∗ ∈ E1∗ and η ∈ A, ϕ1 (v ∗ , w0∗ , η0 ) = ϕ1 (v ∗ , w∗ , η).
The previous lemma defines a mapping φ1 : E1∗ → E1∗ given by φ1 (v ∗ ) = v˜∗ such that v ∗ [UT u] = v˜∗ (u). It is easy to check that φ1 is a homeomorphism. The following lemma is also a direct application of equation (7) with f ≡ u, a constant function.
Lemma 5.2. Let E be a finite dimensional Hilbert space and v0∗ ∈ E1∗ . If T is a surjective isometry on Lip([0, 1], E) satisfying the property-(*), then ˜ (1) ϕ2 (v ∗ , w∗ , [[η]]) = ϕ2 (v0∗ , w∗ , [[η]]) ∀(v ∗ , w∗ , [[η]]) ∈ E1∗ × E1∗ × A. ˜ (2) ϕ3 (v ∗ , w∗ , [[η]]) = ϕ3 (v0∗ , w∗ , [[η]]), ∀(v ∗ , w∗ , [[η]]) ∈ E1∗ × E1∗ × A.
The next lemma asserts the independence of ϕ3 on the first two components of its domain. the proof is written for a complex Hilbert space.
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
15
Lemma 5.3. Let E be a finite dimensional real or complex Hilbert space, v0∗ and w0∗ in E1∗ . If T is a surjective isometry on Lip([0, 1], E) satisfying the property-(*), then ϕ3 (v ∗ , w∗ , [[η]]) = ϕ3 (v0∗ , w0∗ , [[η]]), for ˜ all (v ∗ , w∗ , [[η]]) ∈ E1∗ × E1∗ × A.
Proof. We first fix η ∈ A. Let w0 and w1 be vectors in E1 such that w1 6= w0 , previous results imply that T ∗ (δ(v0∗ ,w0∗ ,η) ) = δ(˜v0∗ ,w˜0∗ ,˜η0 ) and T ∗ (δ(v0∗ ,w1∗ ,η) ) = δ(˜v0∗ ,w˜1∗ ,˜η1 ) . Equivalently we have
(8)
w0∗ [(T f )β (η)] = w ˜0∗ [f β (˜ η0 )] and w1∗ [(T f )β (η)] = w ˜1∗ [f β (˜ η1 )], ∀ f ∈ Lip([0, 1], E).
Equations in (8) imply that there exists η˜ ∈ A and w ˜ ∈ E1 such that w0∗ − w1∗ w ˜0∗ [f β (˜ η0 )] − w ˜1∗ [f β (˜ η1 )] β (T f ) (η) = =w ˜ ∗ [f β (˜ η )], ∀f ∈ Lip([0, 1], E). ∗ ∗ ∗ ∗ kw0 − w1 k kw0 − w1 k Therefore
(9)
w ˜0∗ [f β (˜ η0 )] − w ˜1∗ [f β (˜ η1 )] = kw0∗ − w1∗ k w ˜ ∗ [f β (˜ η )], ∀f ∈ Lip([0, 1], E).
It follows from (9) that given v ⊥ w ˜0 and v ⊥ w ˜1 then v ⊥ w. ˜ Thus w ˜ = a0 w ˜0 + a1 w ˜1 , for some scalars a0 and a1 . Then (9) becomes
(10)
w ˜0∗ {f β (˜ η0 ) − a¯0 kw0∗ − w1∗ kf β (˜ η )} − w ˜1∗ {f β (˜ η1 ) + a¯1 kw0∗ − w1∗ kf β (˜ η )} = 0.
η ) and f β (˜ η1 ) = −a1 kw0∗ − If {w ˜0 , w ˜1 } is linearly independent, then we have f β (˜ η0 ) = a0 kw0∗ − w1∗ kf β (˜ w1∗ kf β (˜ η ). Given v ∈ E1 and f (x) = x · v, for all x ∈ [0, 1], we have a0 = −a1 . Therefore η˜0 ≈ η˜1 . If {w ˜0 , w ˜1 } is linearly dependent, then there exists θ ∈ [0, 2π) such that w ˜1 = eiθ w ˜0 . Hence (10) can be written as: w ˜0∗ {f β (˜ η0 ) − a¯0 kw0∗ − w1∗ kf β (˜ η ) − eiθ f β (˜ η1 ) − eiθ a¯1 kw0∗ − w1∗ kf β (˜ η )} = 0.
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
For f (x) = x · w ˜0 we conclude that 1 − a¯0 kw0∗ − w1∗ k − eiθ − eiθ a¯1 kw0∗ − w1∗ k = 0. If η˜0 is not related with η˜1 then there exists a real valued Lipschitz function λ on [0, 1] such that λβ (˜ η0 ) 6= λβ (˜ η1 ). Then we set λ1 (x) =
λ(x)−λβ (˜ η0 )x λβ (˜ η1 )−λβ (˜ η0 )
and f = λ1 · w ˜0 to conclude that
−a¯0 kw0∗ − w1∗ k λβ1 (˜ η ) − eiθ − eiθ a¯1 kw0∗ − w1∗ k λβ1 (˜ η ) = (eiθ − 1)λβ1 (˜ η ) − eiθ = 0. Since λ1 is real valued, θ = π. Therefore (10) reduces to w ˜0∗ f β (˜ η0 ) − 2f β (˜ η ) + f β (˜ η1 ) = 0. This would imply that δ(v∗ ,w˜0∗ ,˜η) is not an extreme point, since δ(v∗ ,w˜0∗ ,˜η) =
δ(v∗ ,w˜0∗ ,˜η0 ) + δ(v∗ ,w˜0∗ ,˜η1 ) . 2
This contradiction shows that η˜0 ≈ η˜1 . The continuity established in Proposition 5.1 completes the proof. Remark 5.1. Lemmas 5.1 and 5.3 allow us to simplify our notation as follows: φ1 : E1∗ → E1∗ defined by φ1 (v ∗ ) = ϕ1 (v ∗ , w0∗ , η0 ), and φ3 : A˜ → A˜ defined as φ3 ([[η]]) = ϕ3 (v ∗ , w∗ , [[η]]). Furthermore φ3 is a homeomorphism. We also recall that property-(*) reduces (7) to the simpler equation: (11)
w∗ [(T f )β (η)] = w ˜ ∗ [f β (˜ η )], ∀ f ∈ Lip([0, 1], E).
6. Characterization of a class of surjective isometries on Lip([0, 1], E): The real case In this section we assume that E is a real Hilbert space. We start by recalling that ϕ2 : E1∗ × E1∗ × A˜ → E1∗ given by ϕ2 (v ∗ , w∗ , [[η]]) = w ˜ ∗ implies the following: w∗ ((T f )β (η)) = w ˜ ∗ (f β (˜ η )), ∀f ∈ Lip([0, 1], E) and for some η˜ ∈ cl(A). Lemma 5.2 asserts that ϕ2 is independent of the value of the first component in E1∗ × E1∗ × A˜ hence we simplify notation by setting ϕ2 : E1∗ × A˜ → E1∗ given by ϕ2 (w∗ , [[η]]) = w ˜ ∗ . We now fix η ∈ cl(A) and define [[η]]
ϕ2
[[η]]
: E1∗ → E1∗ given by ϕ2 (w∗ ) = ϕ2 (w∗ , [[η]]) = w ˜∗ .
Lemma 6.1. Let E be a finite dimensional (real or complex) Hilbert space and η ∈ cl(A). If T is a surjective [[η]]
isometry on Lip([0, 1], E) satisfying the property-(*), then ϕ2
is an isometry.
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
17
Proof. Let w0∗ and w1∗ be two distinct functionals on E1∗ then Lemma 5.3 implies that
w0∗ ((T f )β (η)) = w ˜0∗ (f β (˜ η )), and w1∗ ((T f )β (η)) = w ˜1∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1], E).
Therefore w0∗ − w1∗ 1 ((T f )β (η)) = (w ˜∗ − w ˜1∗ )(f β (˜ η )) = w ˜2∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1], E), ∗ ∗ ∗ kw0 − w1 k kw0 − w1∗ k 0 with w ˜2∗ =
1 ˜0∗ kw0∗ −w1∗ k (w
−w ˜1∗ ). In particular for f ∈ Lip([0, 1], E) given by f (x) = x · v with v ∈ E we have 1 (w ˜∗ − w ˜1∗ )(v) = w ˜2∗ (v). kw0∗ − w1∗ k 0
This implies that 1 (w ˜∗ − w ˜1∗ ) = w ˜2∗ . kw0∗ − w1∗ k 0 Therefore kw0∗ − w1∗ k = kw ˜0∗ − w ˜1∗ k.
Theorem 6.1. (cf. [6]) Let E1 and F1 be the unit spheres of real Hilbert spaces E and F , respectively. Let T0 : E1 → F1 be a mapping such that kT0 (x) − T0 (y)k ≤ kx − yk for all x, y ∈ E1 and −T0 (E1 ) ⊆ T0 (E1 ). Then T0 can be extended to a linear isometry from E into F .
[[η]]
We now apply Theorem 6.1 to conclude that ϕ2
is the restriction of a real linear isometry on E ∗ . Thus
there exists an isometry V[[η]] on E such that w ˜ ∗ (v) = w∗ (V[[η]] (v)). The notation V[[η]] emphasizes the dependence on η. We impose an additional constraint on T in order to show that η → V[[η]] is constant.
Definition 6.1. Given x ∈ [0, 1] we say that a surjective isometry T on Lip([0, 1], E) is an Id⊗E−isometry if and only if T leaves invariant the subspace spanned by functions of the form Id ⊗ v, with v ∈ E and Id ⊗ v(x) = x · v, for every x ∈ [0, 1].
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Proposition 6.1. Let E be a finite dimensional real Hilbert space. If T is an Id ⊗ E− isometry on Lip([0, 1], E) satisfying property (*), then there exists an isometry V such that, for every η ∈ cl(A), (T f )β (η) = V f β (˜ η ), ∀f ∈ Lip([0, 1], E).
Proof. It follows from Lemma 6.1 and Ding’s Theorem the existence of an isometry, V[[η]] , for each η ∈ cl(A), such that: (T f )β (η) = V[[η]] f β (˜ η ), ∀f ∈ Lip([0, 1], E). In particular, given f = Id ⊗ v then T (f ) = Id ⊗ u, for some u ∈ E. Therefore we have u = V[[η]] v for every η ∈ cl(A). This shows that V[[η]] is independent of η and completes the proof.
The next corollary follows easily from Proposition 6.1.
Corollary 6.1. Let E be a finite dimensional real Hilbert space. If T is an Id⊗E− isometry on Lip([0, 1], E) satisfying property (*), then T maps constant functions onto constant functions.
Proof. If f is constant then (T f )β (η) is zero, for all η ∈ cl(A). Since T f is differentiable almost everywhere we conclude that (T f )0 is equal to zero almost everywhere and then T f is constant.
The characterization of Id ⊗ E−isometries on Lip([0, 1], E) satisfying the condition (*) reduces to the characterization of scalar valued surjective isometries on [0, 1], Lip([0, 1], R). Proposition 6.1 asserts that for every η ∈ cl(A) there exists η˜ ∈ cl(A) such that (T f )β (η) = V f β (˜ η ), and f ∈ Lip([0, 1], E). We select an orthonormal basis for E, {v1 , · · · , vn } and set wi = V ∗ vi . Hence we have {w1 , · · · , wn } is also an orthonormal basis for E. Given f (x) = λ(x)w1 with λ ∈ Lip([0, 1], R) then T f (x) = λ1 v1 + · · · + λn (x)vn , with λi ∈ Lip([0, 1], R). Therefore (T f )β (η) = λβ1 (η)v1 + · · · + λβn (η)vn = λβ (˜ η )V w1 . This implies that λβ1 (η) = λβ (˜ η ) and λβ2 (η) = · · · = λβn (η) = 0, ∀η. Hence λβ2 (η) = a2 , · · · , λβn (η) = an all constants. We then
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
19
write T (f )(x) = λ1 (x)v1 + a2 v2 + · · · + an vn , ∀x ∈ [0, 1]. Since T is an isometry we have |λ(0)| + L(λ) =
q
|λ1 (0)|2 + a22 + · · · + a2n + l(λ1 ),
and condition (*) implies UT λ(0)w1 = λ1 (0)v1 +a2 v2 +· · ·+an vn and hence |λ(0)| =
p
|λ1 (0)|2 + a22 + · · · + a2n .
Therefore L(λ) = L(λ1 ). We now consider g = f − a2 UT∗ v2 , then T (g)(x) = λ1 (x)v1 + a3 v3 + · · · + an vn . Therefore q
|λ(0)|2 + a22 − 2λ(0)a2 h UT w1 , v2 i =
q q |λ1 (0)|2 + a23 + · · · + a2n = |λ(0)|2 − a22 .
From these equations we derive a2 [a2 −λ(0)h UT w1 , v2 i] = 0, which implies that a2 = 0 or a2 = λ(0)h UT w1 , v2 i. If we repeat the same argument for the function h = f + a2 UT∗ v2 we get q
|λ(0)|2 + a22 + 2λ(0)a2 h UT w1 , v2 i =
q
|λ1 (0)|2 + 2a22 + a23 + · · · + a2n =
q |λ(0)|2 + a22 .
This implies that λ(0)a2 h UT w1 , v2 i = 0, and consequently a2 = 0. Similar reasoning applies to conclude that ak = 0, for every k = 3, · · · n. Hence we have that T (λw1 )(x) = λ1 (x)v1 for every x ∈ [0, 1], with |λ1 (0)| = |λ(0)| and L(λ) = L(λ1 ).
Remark 6.1. We observe that T induces an isometry T0 on the subspace of Lip([0, 1], R) consisting of all Lipschitz functions that vanish at zero and equipped with max norm, i.e. kf km = max{kf k∞ , L(f )}, given by T0 (f ) = T (f ). We denote this subspace by Lip0→0 ([0, 1], R). Thus the max norm and the norm considered in this paper coincide when restricted to functions in Lip0→0 ([0, 1], R), since L(f ) ≥ kf k∞ , for every f ∈ Lip0→0 ([0, 1], R).
These considerations legitimize our next constraints on T which allows us to write T as an affine weighted composition operator. In particular, we assume that there exist u and v in E1 such that T (Id2 ⊗v) = Id2 ⊗u.
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
As before, Id2 ⊗ v is a map in Lip([0, 1], E) such that (Id2 ⊗ v)(x) = x2 · v, for every x ∈ [0, 1]. A similar condition has been also considered by Koshimizu in [12]. Therefore Proposition 6.1 implies that τ (η) = τ (˜ η ), ∀η ∈ cl(A).
(12)
Under these constraints on T we define an interval map ψ : [0, 1] → [0, 1] as follows: ψ(x) = τ (φ3 ([[ηx ]])), ∀x ∈ [0, 1], where ηx is a point in cl(A) such that τ (ηx ) = x. Equation (12) implies that ψ(x) = x, for all x ∈ [0, 1], and u = V v.
Theorem 6.2. Let E be a finite dimensional real Hilbert space, u and v unit vectors in E. If T is a Id ⊗ E−isometry on Lip([0, 1], E) satisfying the property-(*) and such that T (Id2 ⊗ v) = Id2 ⊗ u, then there exists V a real isometry such that T (f ) = UT (f (0)) + V (f (x) − f (0)), ∀f ∈ Lip([0, 1], E). Proof. For f ∈ Lip([0, 1], E), f is differentiable almost everywhere then we have (T f )(x) =
Rx
=
Rx
=
V [f (x) − f (0)] + UT (f (0)).
0
0
(T f )0 (t) dt + (T f )(0) V f 0 (t) dt + (T f )(0)
However if we assume that T (Id2 ⊗ v) = (2 Id − Id2 ) ⊗ u, a similar reasoning implies the following result.
Theorem 6.3. Let E be a finite dimensional real Hilbert space, u and v unit vectors in E. If T is a Id ⊗ E−isometry on Lip([0, 1], E) satisfying the property-(*) and such that T (Id2 ⊗ v) = (2 Id − Id2 ) ⊗ u, then there exists V a real isometry such that T (f ) = UT (f (0)) + V (f (1) − f (1 − x)), ∀f ∈ Lip([0, 1], E).
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21
6.1. Characterization of surjective isometries on Lip([0, 1], E): The complex case. In this section we present the complex case. The range space E is a complex finite dimensional Hilbert space, with inner product h , iC . We observe that E is a real Hilbert space when E is equipped with h , iR given by
h u, v iR = Reh u, v iC .
We notice that h u, u iR = h u, u iC , for every u ∈ E. We state theorems 6.2 and 6.3 for this more general case. We recall that property (*) asserts that there exists a complex isometry UT such that T f (0) = UT f (0), for all f ∈ Lip([0, 1], E).
Theorem 6.4. Let E be a finite dimensional complex Hilbert space, u and v unit vectors in E. Let T be a Id ⊗ E−isometry on Lip([0, 1], E) satisfying the property (*). Then (1) If T (Id2 ⊗ v) = Id2 ⊗ u, then there exists V a complex isometry such that
T (f )(x) = UT (f (0)) + V (f (x) − f (0)), ∀f ∈ Lip([0, 1], E).
(2) If T (Id2 ⊗ v) = (2 Id − Id2 ) ⊗ u, then there exists V a complex isometry such that
T (f )(x) = UT (f (0)) + V (f (1) − f (1 − x)), ∀f ∈ Lip([0, 1], E).
Proof. Since T is complex homogeneous then it is also real homogeneous, therefore theorems 6.2 and 6.3 imply the existence of UT and V real isometries such that
T (f )(x) = UT (f (0)) + V (f (x) − f (0)), or T (f )(x) = UT (f (0)) + V (f (1) − f (1 − x)),
for all f ∈ Lip([0, 1], E), and x ∈ [0, 1]. The space E is now considered a real Hilbert space relative to the inner product h , iR . Since T is complex homogeneous we have that T (if ) = iT (f ) hence
UT (if (0)) + V (if (x) − if (0)) = i [UT (f (0)) + V (f (x) − f (0))] ,
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
or UT (if (0)) + V (if (1) − if (1 − x)) = i [UT (f (1)) + V (f (1) − f (1 − x))] , respectively. Convenient choices of Lipschitz functions: f constant and f of the form Id ⊗ v yield UT and V are complex homogeneous therefore isometries on E while equipped with the original inner product h , iC . This completes the proof.
An operator of the form described in the Theorem 6.4 is in fact a surjective isometry. It is easy to check that T is an isometry
kT f k = kUT f (0)k + sup x6=y
kV f (x) − V f (y)k = kf (0)k + L(f ). |x − y|
Furthermore, given g ∈ Lip([0, 1], E) we define f as follows: f (x) = V −1 (g(x) − g(0)) + UT−1 g(0)), for T as in Theorem 6.4-(1) or f (x) = V −1 (g(1) − g(1 − x)) + UT−1 g(0)) for T as in Theorem 6.4-(2). This shows that an operator T of the form described in the Theorem 6.4-(1)and -(2) is a surjective isometry. If also follows from the form established in the Theorem 6.4-(1) that for every f ∈ Lip([0, 1], C) with f (0) = 0 we have T (f ⊗v)(x) = f ⊗V v, where v ∈ E and f ⊗v is a function in Lip([0, 1], E) defined by f ⊗v(x) = f (x)v. As a consequence we have the following result.
Corollary 6.2. Let E be a finite dimensional complex Hilbert space, u and v unit vectors in E. Let T be a surjective isometry on Lip([0, 1], E) satisfying the property (*). Then the following statements are equivalent: (1) T is an Id ⊗ E isometry such that T (Id2 ⊗ v) = Id2 ⊗ u. (2) T (f )(x) = UT (f (0)) + V (f (x) − f (0)), for all f ∈ Lip([0, 1], E) with V a surjective isometry on E. (3) T (f ⊗ v) = f ⊗ V v for every f ∈ Lip([0, 1], C) such that f (0) = 0.
We also observe that a similar result holds for surjective isometries T of the form described in Theorem 6.4-(2).
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
23
Remark 6.2. In general if X is a compact metric space, E is a complex Banach space and ϕ : X → X is an isometry then for any pair of invertible isometries U and V on E it is clear that T given by
(T f )(x) = U f (0) + V [f (ϕ(x) − f (ϕ(0))]
is an invertible isometry on Lip(X, E). It is then tempting to conjecture that all surjective isometries on Lip(X, E), equipped with the norm kf k = kf (0)k + L(f ), are of this form. It is unclear to the authors whether there exist surjective isometries on this setting failing property (*).
References [1] Arens,R. and Kelley, J.,Characterization of spaces of continuous functions over compact Hausdorff spaces,Trans. AMS 62(1947), 499-508. [2] Banach,S., Theorie des operations lineares,Chelsea, Warsaw, 1932. [3] Botelho,F., Fleming, R. and Jamison,J., Extreme points and isometries on vector valued Lipschitz spaces, JMAA (2011) in press. [4] Cambern,M., Isometries of Certain Banach Algebras, Studia Mathematica XXV(1965), 217-225. [5] Cambern,M. and Pathak,V.D., Isometries of Spaces of Differentiable Functions, Math. Japonica 26:3(1981), 253-260. [6] Ding,G.,The 1-Lipschitz mappings between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space, Sci. China Ser. A 45:4(2001), 479-483. [7] Dunford,N. and Schwartz,J.T., Linear Operators Part I: General Theory, (1958) Interscience Oublishers, Inc. New York [8] Fleming,R. and Jamison,J., Isometries on Banach Spaces: Function Spaces, (2003) Chapman & Hall. [9] Fleming,R. and Jamison,J., Isometries on Banach Spaces, (2008) Chapman & Hall. [10] Jim´ enez-Vargas, A. and Villegas-Vallecillos, M., Into linear isometries between spaces of Lipschitz functions, Houston Journal Math. 34, (2008), 1165-1184 [11] Jim´ enez-Vargas, A. and Villegas-Vallecillos, M.,Linear isometries between spaces of vector-valued Lipchitz functions, Proceedings AMS 137:4(2009), 1381-1388. [12] Koshimizu, H.,Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions, preprint (2010). [13] Rao,N.V. and Roy,A.K., Linear isometries on some function spaces, Pacific Journal of Mathematics 38:1(1971),177–192. [14] Stone,M., A general Theory of Spectra, II, Proc. Nat. Acad. Sci. U.S.A.27(1941),83–87.
24
FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected]
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected]
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected]
Abstract. This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.
1. Introduction We consider the space of Lipschitz functions on the interval [0, 1] with values in a finite dimensional Hilbert space E. Given a Lipschitz function f, its Lipschitz constant L(f ) is defined to be L(f ) = supx6=y
kf (x) − f (y)k . |x − y|
In this paper we denote by Lip([0, 1], E) the set of all Lipschitz functions on [0, 1] with values in E, equipped with the norm kf k = kf (0)k + L(f ). We study surjective isometries T on Lip([0, 1], E) with the property that there exists a unitary operator UT , depending on T , such that T f (0) = UT (f (0)), ∀ f ∈ Lip([0, 1], E). We show that under some constraints on the surjective isometry T there exists a unitary operator on E, VT and a surjective isometry ψ on [0, 1] such that T has the representation (1)
T f (x) = UT (f (0)) + VT [f (ψ(x)) − f (ψ(0))] ,
Date: June 30, 2011 . 2000 Mathematics Subject Classification. Primary 46B04; Secondary 46E40. Key words and phrases. Spaces of vector valued Lipschitz functions; surjective isometries; isometric embedding of a space of Lipschitz functions into C(Ω); extreme points of the unit ball of the dual of vector valued Lipschitz function spaces. The research of Bentuo Zheng is partially surported by NSF DMS-1068838 and NSF DMS-0800061. 1
2
FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
for every f ∈ Lip([0, 1], E). We note that there are only two surjective isometries on [0, 1], the identity and the reflection. Furthermore,operators of the form described in (1), acting on Lip(X, E) with X a compact metric space, E an arbitrary Banach space and UT and VT isometries on E, define a group of isometries under composition. This is a subgroup of the isometry group of Lip(X, E). The isometries supported by a Banach space reflect important geometric properties of the space, as a result a characterization of its isometry group is of great interest. Such study is traced back to the pioneering work of Banach and Stone, who obtained the first characterization for the surjective isometries between spaces of scalar valued continuous functions. This well-known result is referred to as the Banach-Stone theorem, see [2, 14]. Researchers have derived extensions of the Banach-Stone theorem for several different settings. We refer the reader to [8] and [9] for a survey of the topic. Cambern and Pathak in [4] and [5], have considered isometries on spaces of scalar valued differentiable functions and determined a representation for the surjective isometries supported by such spaces. Rao and Roy in [13] described the isometries of Lipschitz spaces of scalar valued functions with a variety of different norms. Vargas and Vallecillos in [11] have also considered isometries of spaces of vector valued Lipschitz functions from a compact metric space with values on a strictly convex Banach space, equipped with the norm kf k = max{L(f ), kf k∞ }, see also [10]. In their work, Vargas and Vallecillos require the isometries to satisfy certain natural conditions, similar constraints are also required in our work. The authors in [3] gave a similar representation to the one in [11] for surjective isometries of spaces of vector valued Lipschitz functions from a compact metric space with values on quasi sub-reflexive Banach space with trivial centralizer. In this paper, we extend the results of Rao and Roy in [13] to the vector valued Lipschitz function spaces. Representations for surjective isometries on certain Banach spaces are often derived from the action of their adjoints on the set of the extreme points of the unit ball of the corresponding dual space. A complete description of the extreme points of the dual ball of a Banach space is often very hard to obtain. However a characterization of the extreme points of the dual unit ball of closed subspaces of continuous
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
3
functions was found by Arens and Kelley. This result is a crucial tool in the characterization of the surjective isometries on function spaces. The result of Arens and Kelley is stated in the following theorem.
Theorem 1.1. (cf. [7] pg.441, see also [1]) If X is a closed subspace of C(Ω), which consists of all real or complex valued continuous functions defined on the compact Hausdorff space Ω, and for each ω ∈ Ω, δω ∈ X ∗ is given by δω (f ) = f (ω), f ∈ X , then every extreme point of the unit ball in X ∗ is of the form λδω , with |λ| = 1. If X = C(Ω) the converse also holds, i.e. every element of the form λδω , with |λ| = 1 and ω ∈ Ω, is an extreme point of the unit ball in X ∗ .
The standard approach to the study of the isometries of a function space relies on an isometric embedding of the function space into C(Ω), with Ω a compact Hausdorff space. We adapt this technique in order to embed Lip([0, 1], E) into C(Ω), for a convenient compact and Hausdorff space Ω. Then we proceed with the analysis of the extreme points of Lip([0, 1], E) now identified with a closed subspace of C(Ω). This study leads to natural constraints on a surjective isometry T which allows a representation for T , as an affine weighted composition operators.
2. An isometric embedding of Lip([0, 1], E) into C(Ω) In this section we construct an isometric embedding of Lip([0, 1], E) (with E a finite dimensional complex Banach space) into a space of scalar valued continuous functions on a compact topological space. We denote ˜ = [0, 1] × [0, 1] \ {(x, y) : y ≥ x} by E1∗ the unit ball of the dual space of E with the norm topology. We set X ˜ the Stone-Cech ˇ ˜ Given a function f ∈ Lip([0, 1], E) let f˜ : X ˜ →E and denote by β(X) compactification of X. be defined by f˜(x, y) =
f (x)−f (y) . x−y
˜ Then f β denotes the unique continuous extension of f˜ to β(X).
Lemma 2.1. Let E be a finite dimensional Banach space. Then Lip([0, 1], E) is isometric to a closed ˜ subspace of C(E1∗ × E1∗ × β(X)).
4
FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
˜ as follows: Proof. We define ı : Lip([0, 1], E) → C(E1∗ × E1∗ × β(X)) ˜ ı(f ) = F with F (v ∗ , w∗ , η) = v ∗ (f (0)) + w∗ (f β (η)), ∀(v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X). We show that ı is an isometric isomorphism. The linearity follows from the uniqueness of the extension of ˜ We now prove that ı is an isometry, i.e. for every f ∈ Lip([0, 1], E) f˜ to β(X). ∗ ∗ β kf k = kf (0)k + L(f ) = max(v∗ ,w∗ ,η)∈E ∗ ×E ∗ ×β(X) ˜ |v (f (0)) + w (f (η))|. 1
1
If f (0) = 0, let v ∗ be any functional in E1∗ . If f (0) 6= 0, let v ∗ be a norm 1 functional on E such that ˜ such that v ∗ (f (0)) = kf (0)k. Let {(xα , yα )} be a net in X
L(f ) = lim α
kf (xα ) − f (yα )k . xα − yα
˜ Let η = limα {(xα , yα )}. For each α, we define wα∗ ∈ E ∗ The net {(xα , yα )} has a converging subnet in β(X). 1 to be a norm 1 functional on E such that wα∗
f (xα ) − f (yα ) xα − yα
=
kf (xα ) − f (yα )k . xα − yα
The net of functionals {wα∗ } has a subnet that converges to w0∗ in E1∗ . Then kf k = v ∗ (f (0)) + w0∗ (f β (η)) = kı(f )k∞ . The Closed Graph Theorem asserts that the range of ı is a closed subspace of C(Ω). This concludes the proof.
We denote by R the range of ı, and we observe that ı induces the following isometry between the corresponding dual spaces: ı∗ : R∗ F∗ defined by ı∗ (F ∗ )(g) = F ∗ (ı(g)).
→ Lip([0, 1], E)∗ ı∗ (F ∗ )
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
5
The isometry ı∗ maps the extreme points of R∗1 , the unit ball in R∗ onto the extreme points of the unit ball in the dual space of Lip([0, 1], E). We denote the set of all extreme points of a convex set A by ext(A).
3. Existence of extreme points of Lip([0, 1], E)∗1 In this section we study the extreme points of the unit ball of the dual of Lip([0, 1], E). We identify Lip([0, 1], E) with its isometric image under ı, i.e. R = ı(Lip([0, 1], E)). Theorem 1.1 asserts that ˜ ∗1 ) = {δω : ω ∈ E1∗ × E1∗ × β(X), ˜ with δω (g) = g(ω), for all g}, ext(C(E1∗ × E1∗ × β(X)) ˜ hence the extreme points of R∗1 are functionals of the form δ(v∗ ,w∗ ,η) , with (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X), ext(R∗1 ) ⊆ {δ(v∗ ,w∗ ,η) : δ(v∗ ,w∗ ,η) (F ) = v ∗ (f (0)) + w∗ (f β (η)), ∀f ∈ Lip([0, 1], E)}, (F = ı(f )). Our first lemma shows that the interdependence between the extreme points of Lip([0, 1], E)∗1 and the points ˜ in the growth of β(X).
˜ \ X}. ˜ Lemma 3.1. ext(R∗1 ) ⊆ {δ(v∗ ,w∗ ,η) : (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X) ˜ v ∗ and w∗ ∈ E ∗ , we show that δ(v∗ ,w∗ ,(x,y)) ∈ Proof. Let (x, y) ∈ X, / ext(R∗1 ). We recall that δ(v∗ ,w∗ ,(x,y)) (F ) = 1 (y) . We define δ1 and δ2 in R∗ as follows: v ∗ (f (0)) + w∗ f (x)−f x−y ∗
δ1 (F ) = v (f (0)) + 2w
∗
f (x) − f ( x+y 2 ) x−y
! = δ(v∗ ,w∗ ,(x, x+y )) (F ) 2
and ∗
δ2 (F ) = v (f (0)) + 2w We note that δ(v∗ ,w∗ ,(x,y)) = Given f (t) = (t − x) (t −
∗
f ( x+y 2 ) − f (y) x−y
! = δ(v∗ ,w∗ ,( x+y ,y)) (F ). 2
δ1 +δ2 2 .
x+y 2 ) w,
for every t ∈ [0, 1] and w∗ (w) = 1. We have δ1 (F ) =
δ2 (F ) = δ1 (F ) − (x − y). Therefore δ1 6= δ2 .
x2 +xy ∗ v (w) 2
and
6
FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Now we show that kδ1 k ≤ 1. Similar reasoning applies to δ2 . For every f ∈ Lip([0, 1], E), |δ1 (F )| ≤ |v (f (0))| + 2 w∗
f (x) − f ( x+y 2 ) x−y
∗
! ≤ kf (0)k + L(f ) = kf k = kF k∞ .
These considerations show that δ(v∗ ,w∗ ,(x,y)) is not an extreme point of R∗1 .
Lemma 3.2. Let E be a finite dimensional Hilbert space and let δ(v0∗ ,w0∗ ,η0 ) be an extreme point of R∗1 , with ˜ \ X. ˜ Then for every v ∗ and w∗ ∈ E ∗ , δ(v∗ ,w∗ ,η ) is an extreme point of R∗ . v0∗ , w0∗ ∈ E1∗ and η0 ∈ β(X) 1 1 0 Proof. We fix v ∗ and w∗ ∈ E1∗ . Let {u1 , u2 , . . . un } and {u01 , u02 , . . . u0n } be orthonormal bases for E such that u1 = v0 and u01 = v. We define the unitary operator U =
Pn
i=1
ui ⊗ u0i , with ui ⊗ u0i (u) = hu, u0i iui , for every
u ∈ E. We notice that U v = v0 . We first show that δ(v∗ ,w0∗ ,η0 ) is an extreme point of R∗1 . Suppose otherwise, then δ(v∗ ,w0∗ ,η0 ) =
δ1 + δ2 , δ1 , δ2 ∈ R∗1 and δ1 6= δ2 . 2
For j = 1, and 2 we set δj1 : R F
→
C
→ δj (ı(f1 )),
with f1 (x) = f (x) + U ∗ f (0) − f (0), (U ∗ denotes the adjoint of U ). We show that δ11 and δ21 are distinct functionals in R∗1 . This leads to a contradiction, since δ(v0∗ ,w0∗ ,η) =
δ11 +δ21 2 .
We observe that there exists F ∈ R such that δ1 (F ) 6= δ2 (F ). We set g = f + U f (0) − f (0), hence δ11 (ı(g)) = δ1 (F ) 6= δ2 (F ) = δ21 (ı(g)), proving that δ11 6= δ21 . It is straightforward to check that kδj1 k ≤ 1 (j = 1, 2), |δj1 (F )| = δj (ı(f ))| ≤ kı(f )k∞ = kF k∞ = kf k. This contradiction shows that for every v ∗ ∈ E1∗ , δ(v∗ ,w0∗ ,η0 ) is an extreme point of R∗1 . We assume that δ(v∗ ,w0∗ ,η0 ) ∈ ext(R∗1 ) and prove that δ(˜v∗ ,w∗ ,η0 ) is also in ext(R∗1 ), for some v˜∗ ∈ E1∗ conveniently defined. We consider the following two orthonormal bases for E, {u1 , u2 , . . . un } and {u01 , u02 , . . . u0n }
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
such that u1 = w0 and u01 = w and the unitary operator V =
Pn
i=1
7
u0i ⊗ ui . We have that w∗ (u) = w0∗ (V ∗ u)
and we define v˜∗ ∈ E1∗ as follows: v˜∗ (u) = v0∗ (V ∗ u), where V ∗ stands for the adjoint of V . We assume that δ(˜v∗ ,w1∗ ,η0 ) is not an extreme point of R∗1 . Thus there exist distinct functionals in R∗1 , δ1 and δ2 such that
δ(˜v∗ ,w∗ ,η0 ) =
δ1 + δ2 . 2
We define δ˜j (F ) = δj (ı(V f )) (j = 1, 2) and V f (x) = V (f (x)), for every x ∈ [0, 1]. It is straightforward to check that δ(v0∗ ,w0∗ ,η0 ) =
δ˜1 +δ˜2 2 .
This also leads to a contradiction, since δ˜1 6= δ˜2 and kδ˜i k ≤ 1 (i = 1, 2). This
contradiction shows that δ(˜v∗ ,w∗ ,η0 ) is an extreme point of R∗1 , and the first part of the proof implies that δ(v∗ ,w∗ ,η0 ) must also be an extreme point.
Remark 3.1. Since the set of extreme points of R∗1 is nonempty, then there exists a triplet (v0∗ , w0∗ , η0 ) for which δ(v0∗ ,w0∗ ,η0 ) is an extreme point, and an application of Lemma 3.2 implies that ext(R∗1 ) ⊇ {δ(v∗ ,w∗ ,η0 ) : v ∗ , w∗ ∈ E1∗ }.
4. The trace of an extreme point in [0, 1] ˜ Throughout this section we fix v0∗ and w0∗ in E1∗ . We define the following subset of β X
˜ : δ(v∗ ,w∗ ,η) ∈ ext(R∗1 )}. A = {η ∈ β X 0 0 ˜ \X ˜ and an arbitrary net {(xα , yα )} ∈ X ˜ converging to η, it follows from Lemma 3.1 that Given η ∈ β X {xα } and {yα } must converge to the same point x ∈ [0, 1]. We call this point x the trace of δ(v0∗ ,w0∗ ,η) in [0, 1], or the trace of η in [0, 1]. We now define τ : A → [0, 1] given by τ (η) = x, where x is the unique point in [0, 1] determined by η. In this section we investigate properties of τ to be used in forthcoming results. ˜ that yield identical functionals δ(v∗ ,w∗ ,η) , We first characterize those triplets (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × β(X) when restricted to R.
8
FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Given a scalar valued Lipschitz function λ on [0, 1] we denote by λβ the unique continuous extension of λ ˜ We now start with a preliminary result. to β(X).
Lemma 4.1. If η0 and η1 are two distinct points in A such that τ (η0 ) 6= τ (η1 ), then there exists λ ∈ Lip([0, 1], R) such that λβ (η0 ) 6= λβ (η1 ).
Proof. We set x = τ (η0 ) and y = τ (η1 ). We assume that 0 ≤ x < y ≤ 1. We define t for 0 ≤ t ≤ x+y 2 λ(t) = x+y for x+y ≤ t ≤ 1. 2 2 It is easy to see that λβ (η0 ) = 1 and λβ (η1 ) = 0.
˜ with associated Proposition 4.1. Let (v0∗ , w0∗ , η0 ) and (v1∗ , w1∗ , η1 ) be two distinct triplets in E1∗ × E1∗ × β(X) evaluation functionals δ(v0∗ ,w0∗ ,η0 ) and δ(v1∗ ,w1∗ ,η1 ) respectively. Then δ(v0∗ ,w0∗ ,η0 ) (F ) = δ(v1∗ ,w1∗ ,η1 ) (F ), for all F ∈ R if and only if v0∗ = v1∗ , w0∗ = w1∗ and λβ (η0 ) = λβ (η1 ), for every λ ∈ Lip([0, 1], R).
Proof. If δ(v0∗ ,w0∗ ,η0 ) (F ) = δ(v1∗ ,w1∗ ,η1 ) (F ), for all F ∈ R then (2)
v0∗ [f (0)] + w0∗ [f β (η0 )] = v1∗ [f (0)] + w1∗ [f β (η1 )].
We get v0∗ = v1∗ and v0 = v1 by evaluating (2) at constant functions. Therefore (2) reduces to (3)
w0∗ [f β (η0 )] = w1∗ [f β (η1 )], ∀ f ∈ Lip([0, 1], E).
Equation (3) applied to f (x) = x · v, with v ∈ E, implies w0 = w1 . We apply (3) to functions of the form f = λ · w0 , with λ ∈ Lip([0, 1], R) to conclude that λβ (η0 ) = λβ (η1 ), for every λ ∈ Lip([0, 1], R). Since E is finite dimensional, let {vi } be an orthonormal basis for E, then f ∈ Lip([0, 1], E) has the (unique) representation f (x) =
n X i=1
λi (x) vi , with λi ∈ Lip([0, 1], R).
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
9
Therefore f β (η0 ) = f β (η1 ), ∀ f ∈ Lip([0, 1], E) and the reverse implication follows easily. This completes the proof.
Remark 4.1. It follows easily from (3) that τ (η0 ) = τ (η1 ). In fact, if we assume that τ (η0 ) 6= τ (η1 ) Lemma 4.1 implies the existence of λ such that λβ (η0 ) = 1 and λβ (η1 ) = 0. We set f = λv, with v ∈ E. Thus (3) implies that w0∗ (v) = 0, for every v. This contradiction shows that τ (η0 ) = τ (η1 ).
Lemma 4.2. τ (A) is a dense subset of [0, 1].
Proof. Suppose τ (A) is not dense then there exist y ∈ (0, 1) and an open interval I = (y − , y + ) ⊂ (0, 1) such that I
T
τ (A) = ∅. We consider the evaluation functional δ(v0∗ ,w0∗ ,(y+ 4 ,y− 4 )) . This functional is in the
weak*-closure of the convex hull of ext(R∗1 ). We define the following scalar valued Lipschitz function:
α(x) =
0
if 0 ≤ x ≤ y −
x − (y − 2 )
if y −
2
2
≤x≤y+
2
otherwise.
Now we set f (x) = α(x)w0 , with w0∗ (w0 ) = 1. We denote by F the image of f under ı. Therefore |δ(v0∗ ,w0∗ ,(y+ 4 ,y− 4 )) (F )| = |v0∗ (f (0)) + w0∗ (
α(y+ 4 )−α(y− 4 ) /2
w)| = 1.
On the other hand a convex combination of extreme points is of the form
X
λi,j,k δ(vi∗ ,wj∗ ,ηk ) ,
i=1...n1 , j=1...n2 , k=1...n3
with δ(vi∗ ,wj∗ ,ηk ) ∈ ext(R∗1 ) and λi,j,k scalars. Any such convex combination applied to F yields zero. This would contradict the Krein-Milman Theorem and thus τ (A) is dense in [0, 1].
˜ by cl(A). In the next result we denote the closure of A in β(X)
Lemma 4.3. The map τ is continuous and it has a continuous extension to cl(A).
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Proof. Let {ηα }α∈D be a net in A converging to η ∈ A. For each α ∈ D, let τ (ηα ) = xα and τ (η) = x. We show that xα converges to x. We suppose otherwise. Then there exist > 0 and a subnet of {xα }α∈D1 ( with D1 a directed subset of D and cofinal in D) such that, for all α ∈ D1 , xα is in the complement of T (x − , x + ) [0, 1]. We consider the following net of extreme points of R∗1 , {δ(v0∗ ,w0∗ ,ηα ) }α∈D1 . This net converges to δ(v0∗ ,w0∗ ,η) in the weak*-topology, hence we have
lim w0∗ [f β (ηα )] = w0∗ [f β (η)], ∀f ∈ Lip([0, 1], E). α
This equation leads to a contradiction when applied to the function f = λw0 , with λ a real valued Lipschitz T function defined on [0, 1] such that λ restricted to (x − , x + ) [0, 1] is linear with slope 1 and is constant T on the complement of (x − , x + ) [0, 1]. This contradiction proves that τ is continuous. We now show that τ can be continuously extended to cl(A). Let η0 ∈ cl(A) \ A, then there exists a net of points in A converging to η0 , i.e. limα ηα = η0 . As before, we set τ (ηα ) = xα . The net of extreme points {δ(v0∗ ,w0∗ ,ηα ) } converges to δ(v0∗ ,w0∗ ,η0 ) in the weak*-topology. Equivalently we write
lim w0∗ [f β (ηα )] = w0∗ [f β (η0 )], ∀f ∈ Lip([0, 1], E). α
In particular, for f = λ · w0 with λ ∈ Lip([0, 1], R), we conclude that
(4)
lim λβ (ηα ) = λβ (η0 ), ∀λ ∈ Lip([0, 1], R). α
˜ {(xα , y α )}β with limit ηα , i.e. limβ (xα , y α ) = ηα . Lemma 3.1 For each α, there exists a net of points in X, β β β β α asserts that limβ xα β = limβ yβ = xα ∈ [0, 1]. A similar argument used in the first part of this proof also
shows that {xα } converges to some point in [0, 1], say x0 . ˜ then it is the limit of a net in X, ˜ say limγ (cγ , dγ ) = η0 . Without loss of generality we Since η0 ∈ β(X), assume that limγ cγ = a, limγ dγ = b with a 6= b. We first consider a < b, and x0 < a. Equation in (4) leads
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
to an absurd when applied to the function x λ(x) = x0 +a 2
for
0≤x≤
for
x0 +a 2
If x0 = a < b we consider the following function x λ(x) = −x + a + b
11
x0 +a 2
≤ x ≤ 1.
for
0≤x≤
for
a+b 2
a+b 2
≤ x ≤ 1.
Equation (4) also leads to a contradiction. The remaining cases are eliminated similarly, implying that a = b. Therefore τ can be extended to η0 as follows: τ (η0 ) = a. By constructing a Lipschitz function which is a constant on a small neighborhood of x0 and has nontrivial derivative at a, we readily see that a = x0 . The continuity of τ then follows easily.
Remark 4.2. It is important to observe that τ, now extended to cl(A), is surjective. Given x0 ∈ [0, 1] \ τ (A), there exists a net in τ (A) converging to x0 , hence there exists a convergent net {ηα } in A such that τ (limα ηα ) = x0 .
5. A representation for T : The homeomorphism ϕ A surjective isometry T : Lip([0, 1], E) → Lip([0, 1], E) induces an isometric bijection between the extreme points of R∗1 , i.e. T˜∗ (ext(R∗1 )) = ext(R∗1 ). This translates as follows. For every (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × A there exists (˜ v∗ , w ˜ ∗ , η˜) ∈ E1∗ × E1∗ × A such that
(5)
v ∗ (T f (0)) + w∗ ((T f )β (η)) = v˜∗ (f (0)) + w ˜ ∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1, ], E).
We now investigate whether the statement in (5) can be extended to points in E1∗ × E1∗ × cl(A). Given η0 ∈ cl(A), we consider a net in A, {ηα }α such that limα ηα = η0 . Therefore we have v ∗ (T f (0)) + w∗ ((T f )β (ηα )) = v˜α∗ (f (0)) + w ˜α∗ (f β (˜ ηα )), ∀ f ∈ Lip([0, 1, ], E),
12
FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
since T ∗ (δ(v∗ ,w∗ ,ηα ) = δ(˜vα∗ ,w˜α∗ ,˜ηα ) ). We assume that the net of functionals δ(˜vα∗ ,w˜α∗ ,˜ηα ) converges to δ(˜v0∗ ,w˜0∗ ,˜η0 ) , by selecting a convergent subnet. Hence we conclude that lim {v ∗ (T f (0)) + w∗ ((T f )β (ηα ))} = v˜0∗ (f (0)) + w ˜0∗ (f β (˜ η0 )), ∀ f ∈ Lip([0, 1, ], E). α
If another subnet of δ(˜vα∗ ,w˜α∗ ,˜ηα ) converges to a different functional, δ(˜v1∗ ,w˜1∗ ,˜η1 ) with (˜ v1∗ , w ˜1∗ , η˜1 ) ∈ E1∗ × E1∗ × cl(A), then v˜0∗ (f (0)) + w ˜0∗ (f β (˜ η0 )) = v˜1∗ (f (0)) + w ˜1∗ (f β (˜ η1 )), ∀ f ∈ Lip([0, 1, ], E). Proposition 4.1 implies that v0∗ = v1∗ , w0∗ = w1∗ , and for every λ ∈ Lip([0, 1], R), λβ (˜ η0 ) = λβ (˜ η1 ). We then observe that T˜∗ maps extreme points of R∗1 onto extreme points of R∗1 and also point evaluation functionals δ(v∗ ,w∗ ,η) with (v ∗ , w∗ , η) ∈ E1∗ × E1∗ × cl(A) \ A onto point evaluation functionals of the same form. These considerations lead to the following equivalence relation defined on cl(A): η0 ≈ η1 iff δ(v∗ ,w∗ ,η0 ) |R ≡ δ(v∗ ,w∗ ,η1 ) |R , ∀v ∗ and w∗ ∈ E1∗ . We now denote by A˜ the quotient space of cl(A) relative to ≈, i.e. A˜ = cl(A)/ ≈. It follows from Proposition 4.1 that η0 ≈ η1 iff λβ (η0 ) = λβ (η1 ), ∀λ ∈ Lip([0, 1], R). We represent elements in A˜ by [[η]], with η ∈ cl(A). We now consider the mapping ϕ : E1∗ × E1∗ × A˜ → ˜ given by ϕ(v ∗ , w∗ , [[η]]) = (˜ E1∗ × E1∗ × A, v∗ , w ˜ ∗ , [[˜ η ]]) iff v ∗ [T f (0)] + w∗ [(T f )β (η)] = v˜∗ [f (0)] + w ˜ ∗ [f β (˜ η )], for all f ∈ Lip([0, 1], E). Proposition 4.1 also implies that ϕ is well defined. Furthermore, ϕ is a bijection since T is a surjective isometry.
Proposition 5.1. ϕ is a homeomorphism.
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
13
˜ denoted Proof. We start with a net {(vα∗ , wα∗ , [[ηα ]])}α in E1∗ × E1∗ × A˜ converging to a point in E1∗ × E1∗ × A, by (v ∗ , w∗ , [[η]]). The continuity of ϕ then follows provided we show that {ϕ(vα∗ , wα∗ , [[ηα ]])}α converges to ˜ has a convergent subnet. Without loss of generality ϕ(v ∗ , w∗ , [[η]]). The net {(˜ vα∗ , w ˜α∗ , η˜α )} in E1∗ ×E1∗ ×β(X) ˜ we assume that {(˜ vα∗ , w ˜α∗ , η˜α )} converges to (˜ v0∗ , w ˜0∗ , η˜0 ) in E1∗ × E1∗ × β(X). Since ϕ(vα∗ , wα∗ , ηα ) = (˜ vα∗ , w ˜α∗ , η˜α ) we have
(6)
vα∗ [T f (0)] + wα∗ [(T f )β (ηα )] = v˜α∗ [f (0)] + w ˜α∗ [f β (˜ ηα )], ∀ f ∈ Lip([0, 1], E).
Consequently we have lim v˜α∗ [f (0)] + w ˜α∗ [f β (˜ ηα )] = v˜0∗ [f (0)] + w ˜0∗ [f β (˜ η0 )], α
for every f ∈ Lip([0, 1], E), since |˜ vα∗ [f (0)] + w ˜α∗ [f β (˜ ηα )] − v˜0∗ [f (0)] − w ˜0∗ [f β (˜ η0 )]| ≤ kf (0)kk˜ vα∗ − v˜0∗ k + kf β (˜ ηα ) − f β (˜ η0 )k + kw ˜α∗ − w ˜0∗ kkf β (˜ η0 )k.
On the other hand, we also have lim vα∗ [T f (0)] + wα∗ [(T f )β (ηα )] = v ∗ [T f (0)] + w∗ [(T f )β (η)], α
hence (6) implies that v˜0∗ [f (0)] + w ˜0∗ [f β (˜ η0 )] = v ∗ [T f (0)] + w∗ [(T f )β (η)], ∀ f ∈ Lip([0, 1], E). Consequently ϕ(v ∗ , w∗ , [[η]]) = (˜ v0∗ , w ˜0∗ , [[˜ η0 ]]) implying the continuity of ϕ. The continuity of ϕ−1 is a consequence of the surjectivity of the underlying isometry T .
v∗ , w ˜ ∗ , η˜) ∈ We recall that given w∗ ∈ E1∗ and η ∈ A (respectively cl(A) \ A), for every v ∗ ∈ E1∗ there exists (˜ E1∗ × E1∗ × A (respectively E1∗ × E1∗ × cl(A) \ A) such that
(7)
v ∗ (T f (0)) + w∗ ((T f )β (η)) = v˜∗ (f (0)) + w ˜ ∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1, ], E).
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Since T is a surjective isometry satisfying the following property:
(∗) There exists a unitary operator UT s.t. T (f )(0) = UT (f (0)), ∀ f ∈ Lip([0, 1], E), we have that the correspondence assigning v˜∗ to v ∗ is bijective. ˜ ϕi : E ∗ × E ∗ × A → E ∗ , ϕi (v ∗ , w∗ , [[η]]) = The map ϕ defines three natural continuous maps on E1∗ × E1∗ × A, 1 1 1 πi ϕ(v ∗ , w∗ , [[η]]), with i = 1 or 2, and ϕ3 : E1∗ × E1∗ × A˜ → A˜ given by ϕ3 (v ∗ , w∗ , [[η]]) = π3 ϕ1 (v ∗ , w∗ , [[η]]), with πi (i = 1, 2, or 3) denoting the projection onto component i. Equation (7) with f ≡ u, a constant function, yields v ∗ [UT u] = v˜∗ (u). The next lemma follows easily from this observation.
Lemma 5.1. Let E be a finite dimensional Hilbert space, w0∗ ∈ E1∗ and η0 ∈ A. If T is a surjective isometry on Lip([0, 1], E) satisfying property (*), then for every v ∗ , w∗ ∈ E1∗ and η ∈ A, ϕ1 (v ∗ , w0∗ , η0 ) = ϕ1 (v ∗ , w∗ , η).
The previous lemma defines a mapping φ1 : E1∗ → E1∗ given by φ1 (v ∗ ) = v˜∗ such that v ∗ [UT u] = v˜∗ (u). It is easy to check that φ1 is a homeomorphism. The following lemma is also a direct application of equation (7) with f ≡ u, a constant function.
Lemma 5.2. Let E be a finite dimensional Hilbert space and v0∗ ∈ E1∗ . If T is a surjective isometry on Lip([0, 1], E) satisfying the property-(*), then ˜ (1) ϕ2 (v ∗ , w∗ , [[η]]) = ϕ2 (v0∗ , w∗ , [[η]]) ∀(v ∗ , w∗ , [[η]]) ∈ E1∗ × E1∗ × A. ˜ (2) ϕ3 (v ∗ , w∗ , [[η]]) = ϕ3 (v0∗ , w∗ , [[η]]), ∀(v ∗ , w∗ , [[η]]) ∈ E1∗ × E1∗ × A.
The next lemma asserts the independence of ϕ3 on the first two components of its domain. the proof is written for a complex Hilbert space.
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
15
Lemma 5.3. Let E be a finite dimensional real or complex Hilbert space, v0∗ and w0∗ in E1∗ . If T is a surjective isometry on Lip([0, 1], E) satisfying the property-(*), then ϕ3 (v ∗ , w∗ , [[η]]) = ϕ3 (v0∗ , w0∗ , [[η]]), for ˜ all (v ∗ , w∗ , [[η]]) ∈ E1∗ × E1∗ × A.
Proof. We first fix η ∈ A. Let w0 and w1 be vectors in E1 such that w1 6= w0 , previous results imply that T ∗ (δ(v0∗ ,w0∗ ,η) ) = δ(˜v0∗ ,w˜0∗ ,˜η0 ) and T ∗ (δ(v0∗ ,w1∗ ,η) ) = δ(˜v0∗ ,w˜1∗ ,˜η1 ) . Equivalently we have
(8)
w0∗ [(T f )β (η)] = w ˜0∗ [f β (˜ η0 )] and w1∗ [(T f )β (η)] = w ˜1∗ [f β (˜ η1 )], ∀ f ∈ Lip([0, 1], E).
Equations in (8) imply that there exists η˜ ∈ A and w ˜ ∈ E1 such that w0∗ − w1∗ w ˜0∗ [f β (˜ η0 )] − w ˜1∗ [f β (˜ η1 )] β (T f ) (η) = =w ˜ ∗ [f β (˜ η )], ∀f ∈ Lip([0, 1], E). ∗ ∗ ∗ ∗ kw0 − w1 k kw0 − w1 k Therefore
(9)
w ˜0∗ [f β (˜ η0 )] − w ˜1∗ [f β (˜ η1 )] = kw0∗ − w1∗ k w ˜ ∗ [f β (˜ η )], ∀f ∈ Lip([0, 1], E).
It follows from (9) that given v ⊥ w ˜0 and v ⊥ w ˜1 then v ⊥ w. ˜ Thus w ˜ = a0 w ˜0 + a1 w ˜1 , for some scalars a0 and a1 . Then (9) becomes
(10)
w ˜0∗ {f β (˜ η0 ) − a¯0 kw0∗ − w1∗ kf β (˜ η )} − w ˜1∗ {f β (˜ η1 ) + a¯1 kw0∗ − w1∗ kf β (˜ η )} = 0.
η ) and f β (˜ η1 ) = −a1 kw0∗ − If {w ˜0 , w ˜1 } is linearly independent, then we have f β (˜ η0 ) = a0 kw0∗ − w1∗ kf β (˜ w1∗ kf β (˜ η ). Given v ∈ E1 and f (x) = x · v, for all x ∈ [0, 1], we have a0 = −a1 . Therefore η˜0 ≈ η˜1 . If {w ˜0 , w ˜1 } is linearly dependent, then there exists θ ∈ [0, 2π) such that w ˜1 = eiθ w ˜0 . Hence (10) can be written as: w ˜0∗ {f β (˜ η0 ) − a¯0 kw0∗ − w1∗ kf β (˜ η ) − eiθ f β (˜ η1 ) − eiθ a¯1 kw0∗ − w1∗ kf β (˜ η )} = 0.
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
For f (x) = x · w ˜0 we conclude that 1 − a¯0 kw0∗ − w1∗ k − eiθ − eiθ a¯1 kw0∗ − w1∗ k = 0. If η˜0 is not related with η˜1 then there exists a real valued Lipschitz function λ on [0, 1] such that λβ (˜ η0 ) 6= λβ (˜ η1 ). Then we set λ1 (x) =
λ(x)−λβ (˜ η0 )x λβ (˜ η1 )−λβ (˜ η0 )
and f = λ1 · w ˜0 to conclude that
−a¯0 kw0∗ − w1∗ k λβ1 (˜ η ) − eiθ − eiθ a¯1 kw0∗ − w1∗ k λβ1 (˜ η ) = (eiθ − 1)λβ1 (˜ η ) − eiθ = 0. Since λ1 is real valued, θ = π. Therefore (10) reduces to w ˜0∗ f β (˜ η0 ) − 2f β (˜ η ) + f β (˜ η1 ) = 0. This would imply that δ(v∗ ,w˜0∗ ,˜η) is not an extreme point, since δ(v∗ ,w˜0∗ ,˜η) =
δ(v∗ ,w˜0∗ ,˜η0 ) + δ(v∗ ,w˜0∗ ,˜η1 ) . 2
This contradiction shows that η˜0 ≈ η˜1 . The continuity established in Proposition 5.1 completes the proof. Remark 5.1. Lemmas 5.1 and 5.3 allow us to simplify our notation as follows: φ1 : E1∗ → E1∗ defined by φ1 (v ∗ ) = ϕ1 (v ∗ , w0∗ , η0 ), and φ3 : A˜ → A˜ defined as φ3 ([[η]]) = ϕ3 (v ∗ , w∗ , [[η]]). Furthermore φ3 is a homeomorphism. We also recall that property-(*) reduces (7) to the simpler equation: (11)
w∗ [(T f )β (η)] = w ˜ ∗ [f β (˜ η )], ∀ f ∈ Lip([0, 1], E).
6. Characterization of a class of surjective isometries on Lip([0, 1], E): The real case In this section we assume that E is a real Hilbert space. We start by recalling that ϕ2 : E1∗ × E1∗ × A˜ → E1∗ given by ϕ2 (v ∗ , w∗ , [[η]]) = w ˜ ∗ implies the following: w∗ ((T f )β (η)) = w ˜ ∗ (f β (˜ η )), ∀f ∈ Lip([0, 1], E) and for some η˜ ∈ cl(A). Lemma 5.2 asserts that ϕ2 is independent of the value of the first component in E1∗ × E1∗ × A˜ hence we simplify notation by setting ϕ2 : E1∗ × A˜ → E1∗ given by ϕ2 (w∗ , [[η]]) = w ˜ ∗ . We now fix η ∈ cl(A) and define [[η]]
ϕ2
[[η]]
: E1∗ → E1∗ given by ϕ2 (w∗ ) = ϕ2 (w∗ , [[η]]) = w ˜∗ .
Lemma 6.1. Let E be a finite dimensional (real or complex) Hilbert space and η ∈ cl(A). If T is a surjective [[η]]
isometry on Lip([0, 1], E) satisfying the property-(*), then ϕ2
is an isometry.
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
17
Proof. Let w0∗ and w1∗ be two distinct functionals on E1∗ then Lemma 5.3 implies that
w0∗ ((T f )β (η)) = w ˜0∗ (f β (˜ η )), and w1∗ ((T f )β (η)) = w ˜1∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1], E).
Therefore w0∗ − w1∗ 1 ((T f )β (η)) = (w ˜∗ − w ˜1∗ )(f β (˜ η )) = w ˜2∗ (f β (˜ η )), ∀ f ∈ Lip([0, 1], E), ∗ ∗ ∗ kw0 − w1 k kw0 − w1∗ k 0 with w ˜2∗ =
1 ˜0∗ kw0∗ −w1∗ k (w
−w ˜1∗ ). In particular for f ∈ Lip([0, 1], E) given by f (x) = x · v with v ∈ E we have 1 (w ˜∗ − w ˜1∗ )(v) = w ˜2∗ (v). kw0∗ − w1∗ k 0
This implies that 1 (w ˜∗ − w ˜1∗ ) = w ˜2∗ . kw0∗ − w1∗ k 0 Therefore kw0∗ − w1∗ k = kw ˜0∗ − w ˜1∗ k.
Theorem 6.1. (cf. [6]) Let E1 and F1 be the unit spheres of real Hilbert spaces E and F , respectively. Let T0 : E1 → F1 be a mapping such that kT0 (x) − T0 (y)k ≤ kx − yk for all x, y ∈ E1 and −T0 (E1 ) ⊆ T0 (E1 ). Then T0 can be extended to a linear isometry from E into F .
[[η]]
We now apply Theorem 6.1 to conclude that ϕ2
is the restriction of a real linear isometry on E ∗ . Thus
there exists an isometry V[[η]] on E such that w ˜ ∗ (v) = w∗ (V[[η]] (v)). The notation V[[η]] emphasizes the dependence on η. We impose an additional constraint on T in order to show that η → V[[η]] is constant.
Definition 6.1. Given x ∈ [0, 1] we say that a surjective isometry T on Lip([0, 1], E) is an Id⊗E−isometry if and only if T leaves invariant the subspace spanned by functions of the form Id ⊗ v, with v ∈ E and Id ⊗ v(x) = x · v, for every x ∈ [0, 1].
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Proposition 6.1. Let E be a finite dimensional real Hilbert space. If T is an Id ⊗ E− isometry on Lip([0, 1], E) satisfying property (*), then there exists an isometry V such that, for every η ∈ cl(A), (T f )β (η) = V f β (˜ η ), ∀f ∈ Lip([0, 1], E).
Proof. It follows from Lemma 6.1 and Ding’s Theorem the existence of an isometry, V[[η]] , for each η ∈ cl(A), such that: (T f )β (η) = V[[η]] f β (˜ η ), ∀f ∈ Lip([0, 1], E). In particular, given f = Id ⊗ v then T (f ) = Id ⊗ u, for some u ∈ E. Therefore we have u = V[[η]] v for every η ∈ cl(A). This shows that V[[η]] is independent of η and completes the proof.
The next corollary follows easily from Proposition 6.1.
Corollary 6.1. Let E be a finite dimensional real Hilbert space. If T is an Id⊗E− isometry on Lip([0, 1], E) satisfying property (*), then T maps constant functions onto constant functions.
Proof. If f is constant then (T f )β (η) is zero, for all η ∈ cl(A). Since T f is differentiable almost everywhere we conclude that (T f )0 is equal to zero almost everywhere and then T f is constant.
The characterization of Id ⊗ E−isometries on Lip([0, 1], E) satisfying the condition (*) reduces to the characterization of scalar valued surjective isometries on [0, 1], Lip([0, 1], R). Proposition 6.1 asserts that for every η ∈ cl(A) there exists η˜ ∈ cl(A) such that (T f )β (η) = V f β (˜ η ), and f ∈ Lip([0, 1], E). We select an orthonormal basis for E, {v1 , · · · , vn } and set wi = V ∗ vi . Hence we have {w1 , · · · , wn } is also an orthonormal basis for E. Given f (x) = λ(x)w1 with λ ∈ Lip([0, 1], R) then T f (x) = λ1 v1 + · · · + λn (x)vn , with λi ∈ Lip([0, 1], R). Therefore (T f )β (η) = λβ1 (η)v1 + · · · + λβn (η)vn = λβ (˜ η )V w1 . This implies that λβ1 (η) = λβ (˜ η ) and λβ2 (η) = · · · = λβn (η) = 0, ∀η. Hence λβ2 (η) = a2 , · · · , λβn (η) = an all constants. We then
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
19
write T (f )(x) = λ1 (x)v1 + a2 v2 + · · · + an vn , ∀x ∈ [0, 1]. Since T is an isometry we have |λ(0)| + L(λ) =
q
|λ1 (0)|2 + a22 + · · · + a2n + l(λ1 ),
and condition (*) implies UT λ(0)w1 = λ1 (0)v1 +a2 v2 +· · ·+an vn and hence |λ(0)| =
p
|λ1 (0)|2 + a22 + · · · + a2n .
Therefore L(λ) = L(λ1 ). We now consider g = f − a2 UT∗ v2 , then T (g)(x) = λ1 (x)v1 + a3 v3 + · · · + an vn . Therefore q
|λ(0)|2 + a22 − 2λ(0)a2 h UT w1 , v2 i =
q q |λ1 (0)|2 + a23 + · · · + a2n = |λ(0)|2 − a22 .
From these equations we derive a2 [a2 −λ(0)h UT w1 , v2 i] = 0, which implies that a2 = 0 or a2 = λ(0)h UT w1 , v2 i. If we repeat the same argument for the function h = f + a2 UT∗ v2 we get q
|λ(0)|2 + a22 + 2λ(0)a2 h UT w1 , v2 i =
q
|λ1 (0)|2 + 2a22 + a23 + · · · + a2n =
q |λ(0)|2 + a22 .
This implies that λ(0)a2 h UT w1 , v2 i = 0, and consequently a2 = 0. Similar reasoning applies to conclude that ak = 0, for every k = 3, · · · n. Hence we have that T (λw1 )(x) = λ1 (x)v1 for every x ∈ [0, 1], with |λ1 (0)| = |λ(0)| and L(λ) = L(λ1 ).
Remark 6.1. We observe that T induces an isometry T0 on the subspace of Lip([0, 1], R) consisting of all Lipschitz functions that vanish at zero and equipped with max norm, i.e. kf km = max{kf k∞ , L(f )}, given by T0 (f ) = T (f ). We denote this subspace by Lip0→0 ([0, 1], R). Thus the max norm and the norm considered in this paper coincide when restricted to functions in Lip0→0 ([0, 1], R), since L(f ) ≥ kf k∞ , for every f ∈ Lip0→0 ([0, 1], R).
These considerations legitimize our next constraints on T which allows us to write T as an affine weighted composition operator. In particular, we assume that there exist u and v in E1 such that T (Id2 ⊗v) = Id2 ⊗u.
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
As before, Id2 ⊗ v is a map in Lip([0, 1], E) such that (Id2 ⊗ v)(x) = x2 · v, for every x ∈ [0, 1]. A similar condition has been also considered by Koshimizu in [12]. Therefore Proposition 6.1 implies that τ (η) = τ (˜ η ), ∀η ∈ cl(A).
(12)
Under these constraints on T we define an interval map ψ : [0, 1] → [0, 1] as follows: ψ(x) = τ (φ3 ([[ηx ]])), ∀x ∈ [0, 1], where ηx is a point in cl(A) such that τ (ηx ) = x. Equation (12) implies that ψ(x) = x, for all x ∈ [0, 1], and u = V v.
Theorem 6.2. Let E be a finite dimensional real Hilbert space, u and v unit vectors in E. If T is a Id ⊗ E−isometry on Lip([0, 1], E) satisfying the property-(*) and such that T (Id2 ⊗ v) = Id2 ⊗ u, then there exists V a real isometry such that T (f ) = UT (f (0)) + V (f (x) − f (0)), ∀f ∈ Lip([0, 1], E). Proof. For f ∈ Lip([0, 1], E), f is differentiable almost everywhere then we have (T f )(x) =
Rx
=
Rx
=
V [f (x) − f (0)] + UT (f (0)).
0
0
(T f )0 (t) dt + (T f )(0) V f 0 (t) dt + (T f )(0)
However if we assume that T (Id2 ⊗ v) = (2 Id − Id2 ) ⊗ u, a similar reasoning implies the following result.
Theorem 6.3. Let E be a finite dimensional real Hilbert space, u and v unit vectors in E. If T is a Id ⊗ E−isometry on Lip([0, 1], E) satisfying the property-(*) and such that T (Id2 ⊗ v) = (2 Id − Id2 ) ⊗ u, then there exists V a real isometry such that T (f ) = UT (f (0)) + V (f (1) − f (1 − x)), ∀f ∈ Lip([0, 1], E).
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
21
6.1. Characterization of surjective isometries on Lip([0, 1], E): The complex case. In this section we present the complex case. The range space E is a complex finite dimensional Hilbert space, with inner product h , iC . We observe that E is a real Hilbert space when E is equipped with h , iR given by
h u, v iR = Reh u, v iC .
We notice that h u, u iR = h u, u iC , for every u ∈ E. We state theorems 6.2 and 6.3 for this more general case. We recall that property (*) asserts that there exists a complex isometry UT such that T f (0) = UT f (0), for all f ∈ Lip([0, 1], E).
Theorem 6.4. Let E be a finite dimensional complex Hilbert space, u and v unit vectors in E. Let T be a Id ⊗ E−isometry on Lip([0, 1], E) satisfying the property (*). Then (1) If T (Id2 ⊗ v) = Id2 ⊗ u, then there exists V a complex isometry such that
T (f )(x) = UT (f (0)) + V (f (x) − f (0)), ∀f ∈ Lip([0, 1], E).
(2) If T (Id2 ⊗ v) = (2 Id − Id2 ) ⊗ u, then there exists V a complex isometry such that
T (f )(x) = UT (f (0)) + V (f (1) − f (1 − x)), ∀f ∈ Lip([0, 1], E).
Proof. Since T is complex homogeneous then it is also real homogeneous, therefore theorems 6.2 and 6.3 imply the existence of UT and V real isometries such that
T (f )(x) = UT (f (0)) + V (f (x) − f (0)), or T (f )(x) = UT (f (0)) + V (f (1) − f (1 − x)),
for all f ∈ Lip([0, 1], E), and x ∈ [0, 1]. The space E is now considered a real Hilbert space relative to the inner product h , iR . Since T is complex homogeneous we have that T (if ) = iT (f ) hence
UT (if (0)) + V (if (x) − if (0)) = i [UT (f (0)) + V (f (x) − f (0))] ,
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
or UT (if (0)) + V (if (1) − if (1 − x)) = i [UT (f (1)) + V (f (1) − f (1 − x))] , respectively. Convenient choices of Lipschitz functions: f constant and f of the form Id ⊗ v yield UT and V are complex homogeneous therefore isometries on E while equipped with the original inner product h , iC . This completes the proof.
An operator of the form described in the Theorem 6.4 is in fact a surjective isometry. It is easy to check that T is an isometry
kT f k = kUT f (0)k + sup x6=y
kV f (x) − V f (y)k = kf (0)k + L(f ). |x − y|
Furthermore, given g ∈ Lip([0, 1], E) we define f as follows: f (x) = V −1 (g(x) − g(0)) + UT−1 g(0)), for T as in Theorem 6.4-(1) or f (x) = V −1 (g(1) − g(1 − x)) + UT−1 g(0)) for T as in Theorem 6.4-(2). This shows that an operator T of the form described in the Theorem 6.4-(1)and -(2) is a surjective isometry. If also follows from the form established in the Theorem 6.4-(1) that for every f ∈ Lip([0, 1], C) with f (0) = 0 we have T (f ⊗v)(x) = f ⊗V v, where v ∈ E and f ⊗v is a function in Lip([0, 1], E) defined by f ⊗v(x) = f (x)v. As a consequence we have the following result.
Corollary 6.2. Let E be a finite dimensional complex Hilbert space, u and v unit vectors in E. Let T be a surjective isometry on Lip([0, 1], E) satisfying the property (*). Then the following statements are equivalent: (1) T is an Id ⊗ E isometry such that T (Id2 ⊗ v) = Id2 ⊗ u. (2) T (f )(x) = UT (f (0)) + V (f (x) − f (0)), for all f ∈ Lip([0, 1], E) with V a surjective isometry on E. (3) T (f ⊗ v) = f ⊗ V v for every f ∈ Lip([0, 1], C) such that f (0) = 0.
We also observe that a similar result holds for surjective isometries T of the form described in Theorem 6.4-(2).
ISOMETRIES ON SPACES OF VECTOR VALUED LIPSCHITZ FUNCTIONS
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Remark 6.2. In general if X is a compact metric space, E is a complex Banach space and ϕ : X → X is an isometry then for any pair of invertible isometries U and V on E it is clear that T given by
(T f )(x) = U f (0) + V [f (ϕ(x) − f (ϕ(0))]
is an invertible isometry on Lip(X, E). It is then tempting to conjecture that all surjective isometries on Lip(X, E), equipped with the norm kf k = kf (0)k + L(f ), are of this form. It is unclear to the authors whether there exist surjective isometries on this setting failing property (*).
References [1] Arens,R. and Kelley, J.,Characterization of spaces of continuous functions over compact Hausdorff spaces,Trans. AMS 62(1947), 499-508. [2] Banach,S., Theorie des operations lineares,Chelsea, Warsaw, 1932. [3] Botelho,F., Fleming, R. and Jamison,J., Extreme points and isometries on vector valued Lipschitz spaces, JMAA (2011) in press. [4] Cambern,M., Isometries of Certain Banach Algebras, Studia Mathematica XXV(1965), 217-225. [5] Cambern,M. and Pathak,V.D., Isometries of Spaces of Differentiable Functions, Math. Japonica 26:3(1981), 253-260. [6] Ding,G.,The 1-Lipschitz mappings between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space, Sci. China Ser. A 45:4(2001), 479-483. [7] Dunford,N. and Schwartz,J.T., Linear Operators Part I: General Theory, (1958) Interscience Oublishers, Inc. New York [8] Fleming,R. and Jamison,J., Isometries on Banach Spaces: Function Spaces, (2003) Chapman & Hall. [9] Fleming,R. and Jamison,J., Isometries on Banach Spaces, (2008) Chapman & Hall. [10] Jim´ enez-Vargas, A. and Villegas-Vallecillos, M., Into linear isometries between spaces of Lipschitz functions, Houston Journal Math. 34, (2008), 1165-1184 [11] Jim´ enez-Vargas, A. and Villegas-Vallecillos, M.,Linear isometries between spaces of vector-valued Lipchitz functions, Proceedings AMS 137:4(2009), 1381-1388. [12] Koshimizu, H.,Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions, preprint (2010). [13] Rao,N.V. and Roy,A.K., Linear isometries on some function spaces, Pacific Journal of Mathematics 38:1(1971),177–192. [14] Stone,M., A general Theory of Spectra, II, Proc. Nat. Acad. Sci. U.S.A.27(1941),83–87.
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FERNANDA BOTELHO, JAMES JAMISON, AND BENTUO ZHENG
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected]
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected]
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152 E-mail address: [email protected]
