# Isometries between Spaces of Vector-Valued Differentiable Functions

Authors: Alireza Ranjbar-Motlagh
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This article characterizes the isometries between spaces of all differentiable functions from a compact interval of the real line into a strictly convex Banach space.


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## Body

## 1. Introduction

The main purpose of this article is to characterize the isometries of the space of all (continuously) differentiable functions from a compact interval of the real line into a strictly convex Banach space.Cambern ([1], Theorem 6.5.5) and Pathak [2] investigated the surjective linear isometries over spaces of complex-valued differentiable functions on the interval 0,1 and gave a representation for such operators, and Jarosz and Pathak [3] studied those operators over differentiable function spaces defined on te compact subsets of the real line (without isolated points). Also, Wang [4] studied the isometries between spaces of scalar-valued (real case and complex case) differentiable functions (and vanish at infinity) on the locally compact subsets of the Euclidean spaces (without isolated points) and gave a representation for such operators.On the other hand, Botelho and Jamison [5] extended these results to the surjective linear isometries on spaces of 1-time differentiable functions on 0,1 with values in a finite-dimensional Hilbert space. In [5], the main result is valid whenever the (real) dimension of the Hilbert space is bigger than one. Compare with the assumptions in ([6], Theorem 2.13). In [6], the authors investigated the isometries between spaces of p-times differentiable functions (and vanish at infinity) on an open subset of the real line with values in a strictly convex Banach space. Also, Li and Wang [7] studied the isometries between spaces of p-times differentiable functions (and vanish at infinity) on an open subset of the Euclidean space with values in a reflexive and strictly convex Banach space. In [7], there is a gap in the proof of Theorem 3.5 (page 553); in the proof of Claim 1, why ∂γTgτx:γ∈Γ must have at most one nonzero term? Compare with the proofs in ([4], Lemma 2.5) and ([2], Lemma 2.3).Suppose thatV,· and W,· are Banach spaces (on the real line). Denote the space of all C1-functions f:0,1→V by C10,1,V, and on this space, we consider the following norm:
(1)f1≔maxt∈0,1ft+f′t=maxt∈0,1ft1<∞.LetT:C10,1,V→C10,1,V be a surjective linear isometry. In [5], it is shown when V is a finite dimensional Hilbert space, there exists a linear isometry J:V→V such that either Tft=Jft or Tft=Jf1−t, for all f∈C10,1,V and t∈0,1. In this article, we extend the above result to a surjective isometry T:C10,1,V→C10,1,W, whenever Banach spaces V and W are strictly convex.In [6], the authors described the surjective isometries between spaces of p-times differentiable functions (and vanish at infinity) on an open subset of the real line with values in a strictly convex Banach space (with dimension greater than one). Such a representation is as Tft=Jtf∘τt, where Jt:V→W is a surjective linear isomorphism (for all t∈0,1) and τ is a diffeomorphism (over 0,1), and its proof is based on the main result of [8]. By using the proof of ([6], Lemma 2.12), we recover their representation in our setting. Therefore, our description is better and its proof is somewhat shorter and more elementary than the one in [6]. Also, in the appendix, we provide a proof for Theorem 13 for special case =W=R . In a forthcoming article, we extend our result to the class of p-times differentiable functions.


## 2. Characterization ofT-Set in C10,1,V

Suppose thatV,· is a Banach space on the real line. For v1,v2∈V\0 and x∈0,1, we define
(2)Sx,v1,v2≔f∈C10,10,1,V:f1=fx1&fx∈v1+&f′x∈v2+,where v+≔u∈V:u=λv,forsomeλ≥0.AT-set in a Banach space U is a subset S of U with the property that for any finite collection x1,⋯,xn∈S, ∑inxi=∑inxi, and such that S is maximal with respect to this property.In this section, we show that everyT-set of C10,1,V has a unique representation by some Sx,v1,v2, up to v1+ and v2+, whenever V is strictly convex (compare with ([9], Lemma 7.2.2)). We start this section with two elementary Lemmas.Lemma 1.
For anyx∈0,1 and η>0 such that 0<x−η<x+η<1, there exists ϕ∈C10,1,R in such a way that ϕx=0, suppϕ⊂x−η,x+η and
(3)ϕ1=ϕx1>ϕt1,for all t∈0,1\x. Moreover, when x=0 or x=1, we can state and prove a similar result.Proof.
For0<δ<ε<1/4, define fε,δ:−ε,ε→R as follows:
(4)fε,δt≔t+δ−δift≥0,−−t+δ+δift≤0.Then, it is easy to verify thatfε,δ∈C1−ε,ε,R and
(5)fε,δ1=fε,δ01=0+12δ>fε,δt1,for all t∈−ε,ε\0. It is clear that fε,δ can be extended as a C1-function on −1/2,1/2 with support in −ε+,ε+, for any ε+∈ε,1/2. Denote this extension of fε,δ by f~ε,δ. Now, by choosing δ>0 small enough (fixing ε and ε+), we have
(6)f~ε,δ1=f~ε,δ01>f~ε,δt1,for all t∈−1/2,1/2\0. Now, one can easily construct the desired function ϕ as in Lemma 1.Forv∈V, the function v^∈C10,1,V denotes the constant function with value v.Lemma 2.
LetV be a Banach space, v1∈V and v2∈V\0. For any x∈0,1, there exists ϕ∈C10,1,V in such a way that ϕx=v1, ϕ′x=v2 and
(7)ϕ1=ϕx1>ϕt1,for all t∈0,1\x.Proof.
Letϕ0 be a function with the properties as in Lemma 1. Then, ϕ≔v^1+ϕ0/ϕ0′xv^2 is the desired function.Theorem 3.
LetV be a strictly convex Banach space. Then, every T-set of C10,1,V is as Sx,v1,v2, for some x∈0,1 and v1,v2∈V\0, and vice versa Sx,v1,v2 is a T-set of C10,1,V.Proof.
First, we show thatSx,v1,v2 is a T-set of C10,1,V, for any x∈0,1 and v1,v2∈V\0. To do this, it is enough to show the maximality property for Sx,v1,v2. Suppose that h∈C10,1,V such that h+f1=h1+f1, for any f∈Sx,v1,v2, we show that h∈Sx,v1,v2. Suppose that h1>hx1. So, there exists an open neighborhood of x in 0,1, say Ux, such that h1>ht1, for all t∈Ux. By Lemma 1, there exists f1∈Sx,v1,v2 such that f1x=0, f1′x=v2 and suppf1⊂Ux, and also
(8)f11=f1x1>f1t1,for all t∈0,1\x. Then, we have
(9)ht+f1t1≤ht1+f1t1<h1+f11,for all t∈0,1. This is a contradiction. Therefore, we get h1=hx1.Next, we prove thathx∈v1+ and h′x∈v2+. We know that f2≔v^1+f1∈Sx,v1,v2 (see the proof of Lemma 2) and
(10)ht+f2t1≤ht1+f2t1<h1+f21,for all t∈0,1\x. Also, we have
(11)h+f21=hx+f2x1=hx+f2x+h′x+f2′x≤hx+f2x+h′x+f2′x=hx+h′x+f2x+f2′x≤h1+f21.By Equation (10) and the assumptions, we see that all inequalities in Equation (11) are equalities. This implies that
(12)hx+f2x=hx+f2x,h′x+f2′x=h′x+f2′x.SinceV is strictly convex, this implies that hx∈f2x+ and h′x∈f2′x+ and then hx∈v1+ and h′x∈v2+.To prove the converse, suppose thatS is a T-set in C10,1,V. For f∈S, define
(13)Kf≔t∈0,1:f1=ft1.It is clear thatKf is a compact nonempty subset of 0,1. Also, for any finite collection of elements f1,f2,⋯,fn∈S, we have ⋂i=1nKfi≠∅. By contradiction, suppose that ⋂i=1nKfi=∅. This implies that for any t∈0,1, there exists some fjt, for 1≤jt≤n, such that fjtt1<fjt1. Then, we obtain
(14)∑i=1nfit1≤∑i=1nfit1<∑i=1nfi1,for all t∈0,1. So, we get
(15)∑i=1nfit1<∑i=1nfi1,for all t∈0,1. This is a contradiction. So, ⋂i=1nKfi≠∅. This implies that ⋂f∈SKf≠∅. Let x∈⋂f∈SKf. Finally, by the maximality property of S, since V is strictly convex, we see that S=Sx,v1,v2, for some v1,v2∈V\0.Next, we state a few simple facts aboutT-sets in C10,1,V, whenever V is a strictly convex Banach space.Proposition 4.
LetV be a strictly convex Banach space. Consider the space C10,1,V, then we have
(i)
Sx,v1,v2 is not trivial, i.e., Sx,v1,v2≠0(ii)
Sx,v1,v2∩Sx¯,v¯1,v¯2 is not trivial, whenever x≠x¯(iii)
Ifx≠x¯, or v1∉v¯1+, or v2∉v¯2+, then Sx,v1,v2≠Sx¯,v¯1,v¯2(iv)
Ifv1∉v¯1+, then(16)Sx,v1,v2⋂Sx,v¯1,v2=f∈C10,1,V:fx=0⋂Sx,v1,v2.(v)
Ifv2∉v¯2+, then(17)Sx,v1,v2⋂Sx,v1,v¯2=f∈C10,1,V:f′x=0⋂Sx,v1,v2.(vi)
Ifv1∉v¯1+ and v2∉v¯2+, then(18)Sx,v1,v2⋂Sx,v¯1,v¯2=0.Proof.
It is straightforward (using Lemma1 and Lemma 2).


## 3. Main Results

In this section, by using results of the previous section aboutT-sets in C10,1,V, we obtain a few important properties of a given isometry T:C10,1,V→C10,1,W in order to characterize such isometry.Proposition 5.
LetT:C10,1,V→C10,1,W be a surjective linear isometry, where V and W are two strictly convex Banach spaces. Then,
(i)
T maps a T-set in C10,1,V to a T-set in C10,1,W. In particular, for any x∈0,1 and v1,v2∈V\0, there exist y∈0,1 and w1,w2∈W\0 such that TSx,v1,v2=Sy,w1,w2(ii)
IfTSx,v1,v2=Sy,w1,w2 and TSx,v¯1,v¯2=Sy¯,w¯1,w¯2, then y=y¯Proof.
(i) SinceT is an isometry, by definition, it is easy to see that T maps a T-set to a T-set. Now, by Theorem 3, the proof is complete. (ii) By contradiction, suppose that y≠y¯.First, we assume thatdimV>1. By Proposition 4 (vi), there exist u1,u2∈V\0 such that
(19)Sx,u1,u2⋂Sx,v1,v2=0,Sx,u1,u2⋂Sx,v¯1,v¯2=0.Therefore, we obtain(20)TSx,u1,u2⋂Sy,w1,w2=0,TSx,u1,u2⋂Sy¯,w¯1,w¯2=0.On the other hand, by part (i), we know thatTSx,u1,u2=Sz,ζ1,ζ2, for some z∈0,1 and ζ1,ζ2∈W\0. Also, either z≠y or z≠y¯. This is a contradiction, by Proposition 4 (ii).Next, we assume thatV=R and dimW>1. Then, there exist ζ1,ζ2∈W\0 and x¯∈0,1\x such that TSx¯,v^1,v^2=Sy,ζ1,ζ2, for some v^1,v^2∈V\0. Now, by considering T−1 and since T−1Sy,w1,w2=Sx,v1,v2 and T−1Sy,ζ1,ζ2=Sx¯,v^1,v^2, we obtain a contradiction as before (note that x≠x¯).Finally, we assume thatV=W=R (see [3], page 202). Also, we present a proof for this in the appendix.Corollary 6.
LetT:C10,1,V→C10,1,W be a surjective linear isometry, where V and W are two strictly convex Banach spaces. Then, there exists a bijection ΦT:0,1→0,1 such that TSx,v1,v2=SΦTx,w1,w2, ΦTx does not depend on v1,v2∈V (and w1,w2∈W).Proof.
It is an immediate consequence of Proposition5.Theorem 7.
LetT:C10,1,V→C10,1,W be a surjective linear isometry, where V and W are two strictly convex Banach spaces. Then, T maps constant functions to constant functions. In particular, T induces a linear isometry from V onto W.Proof.
Suppose thatv∈V\0 and the function ΦT is as in Corollary 6. It is clear that v^∈Sx,v,v2, for all x∈0,1 and v2∈V. Therefore, by Proposition 4 (iv) and (v) and Proposition 5 (ii), for any x∈0,1, we see that either Tv^ΦTx=0 or Tv∧′ΦTx=0. Now, the theorem is an immediate consequence of the following simple fact from real analysis:
(i)
Leth:R→V be a differentiable function such that for any x∈R, either hx=0 or h′x=0. Then, h is a constant functionProposition 8.
LetT:C10,1,V→C10,1,W be a surjective linear isometry, where V and W are two strictly convex Banach spaces. Consider x∈0,1 and v1,v2∈V\0 and suppose that TSx,v1,v2=Sy,w1,w2, for some y∈0,1 and w1,w2∈W\0. Then,
(i)
Ifv¯2∈V\0, then,(21)TSx,v1,v¯2=Sy,w1,w¯2,for some w¯2∈W\0(ii)
Ifv¯1∈V\0, then,(22)TSx,v¯1,v2=Sy,w¯1,w2,for some w¯1∈W\0Proof.
(i) By Proposition5 (ii), we know that TSx,v1,v¯2=Sy,w¯1,w¯2, for some w¯1,w¯2∈W\0. On the other hand, the constant function v^1∈C10,1,V belongs to Sx,v1,v, for all v∈V. Therefore, by Theorem 7, Tv^1∈C10,1,W is a constant function with value in w1+∩w¯1+. This implies that w¯1∈w1+ (note that Tv^1≠0). So, this completes the proof. (ii) The proof is an immediate consequence of part (i) applied to T−1 (by contradiction and using Proposition 4 (vi)).Lemma 9.
Suppose thatf∈C20,1,V, i.e.,f is continuously twice differentiable, such that fx=0 and f′x≠0, for some x∈0,1. For any η>0 such that 0<x−η<x+η<1, there exists ϕ∈C10,1,V such that ϕx=0, ϕ′x∈f′x+, suppϕ⊂x−η,x+ηand
(23)ϕ1=ϕx1>ϕt1,ϕ+f1=ϕx+fx1>ϕt+ft1,for all t∈0,1\x. Moreover, when x=0 or x=1, we can state and prove a similar result.Proof.
Sincef∈C20,1,V, there is a positive constant M such that
(24)f′s−f′t≤Ms−t,for all s,t∈0,1. By using this fact, the rest of proof is similar to the proof of Lemma 1, with a slight modification.Remark 10.
Lemma9 is meaningful when f′x=0. In fact, for any v∈V\0, there exists ϕ∈C10,1,V satisfying the conditions in Lemma 9, except, replacing the condition of ϕ′x∈f′x+ with ϕ′x∈v+\0.Theorem 11.
LetT:C10,1,V→C10,1,W be a surjective linear isometry, where V and W are two strictly convex Banach spaces. Suppose that ΦT is as in Corollary 6 and consider x∈0,1. Suppose that f∈C20,1,V satisfies the condition fx=0, then TfΦTx=0.Proof.
By Lemma9 and Remark 10, there exists a function ϕ∈C10,1,V in such a way that ϕx=0:
(25)ϕ1=ϕx1>ϕt1,ϕ+f1=ϕx+fx1>ϕt+ft1,for all t∈0,1\x. This implies that ϕ+f∈Sx,v,v2, for all v∈V\0 and some v2∈V\0. Now, by Proposition 8 (ii), we see that Tϕ+f∈SΦTx,w,w2, for all w∈W\0 and some w2∈W\0. So, by Proposition 4 (iv), we have Tϕ+fΦTx=0. Similarly, we can show that TϕΦTx=0. This completes the proof of theorem.Corollary 12.
In Theorem11, we can replace the condition f∈C20,1,V with the weaker assumption f∈C10,1,V.Proof.
It is an immediate consequence of the density ofC20,1,V in C10,1,V (with ·1 norm).Theorem 13.
LetT:C10,1,V→C10,1,W be a surjective linear isometry, where V and W are two strictly convex Banach spaces. Then, there exists a surjective linear isometry J:V→W, such that either Tft=Jft or Tft=Jf1−t, for all f∈C10,1,V and t∈0,1.Proof.
By Theorem7, we know that T maps the constant function v^∈C10,1,V with value v∈V to the constant function w^∈C10,1,W, for some w∈W. So, T induces a surjective linear isometry J:V→W. Now, let f be an arbitrary element of C10,1,V and let x be an arbitrary element of 0,1. Define g≔f−fx^∈C10,1,V. It is clear that gx=0, so by Corollary 12, we obtain TgΦTx=0. This implies that TfΦTx=Jfx, for all f∈C10,1,V and x∈0,1. Finally, by a standard argument, we can show that ΦT is differentiable on 0,1 and the absolute value of its derivative is a constant function with value 1. This completes the proof of theorem.Remark 14.
Theorem13 remains meaningful and valid for a surjective linear isometry T:C1a,b,V→C1c,d,W as well, and as a consequence, we see that the intervals a,b and c,d should have equal length.



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