A Remark on Isometries of Absolutely Continuous Spaces

Suppose that (V, ‖·‖) and (W, ‖·‖) are Banach spaces (real or complex) and X and Y are compact subsets of the real line (with the induced distance and order from R). A map f: X⟶ V is absolutely continuous, if for every ε> 0, there exists a δ > 0 such that 􏽐ni�1 ‖f(bi) − f(ai)‖< ε, whenever ai, bi 􏼈 􏼉1≤ i≤ n is a (finite) sequence of mutually disjoint intervals in R whose end points are in X and 􏽐ni�1 |bi − ai|< δ. Also, the total variation of f (on X) is defined as


Introduction
Suppose that (V, ‖·‖) and (W, ‖·‖) are Banach spaces (real or complex) and X and Y are compact subsets of the real line (with the induced distance and order from R). A map f: X ⟶ V is absolutely continuous, if for every ε > 0, there exists a δ > 0 such that n i�1 ‖f(b i ) − f(a i )‖ < ε, whenever a i , b i 1 ≤ i ≤ n is a (finite) sequence of mutually disjoint intervals in R whose end points are in X and n i�1 |b i − a i | < δ. Also, the total variation of f (on X) is defined as where the supremum is taken over all (finite) increasing sequences x i 1 ≤ i ≤ n of points in X. e set of all absolutely continuous maps from X into V is denoted by AC(X, V), and on this space, we consider the following norm: where ‖f‖ ∞ denotes the supremum norm of f (on X).
In this article, we study isometries on such function spaces. We attempt to characterize isometries on these function spaces, under certain conditions. Suppose that T: AC(X, V) ⟶ AC(Y, W) surjective linear isometry, where V and W are Banach spaces with trivial (one-dimensional) centralizers. We show that if T and T − 1 have property R, then T is a "weighted composition operator" of the form where ϕ: Y ⟶ X and its inverse are absolutely continuous, and J is an absolutely continuous map from Y into the space of surjective linear isometries from V into W. Compare with [1], eorem 1.8 and [2], eorem 5.3. In particular, we present a direct proof for the main result of [3], which is somewhat shorter and more elementary than the one in [3]. Our proof is based on the principal ideas of [1] which convert the initial function space to the class of continuous maps (equipped with the supremum norm) and then appeal to the classical Banach-Stone theorem.

Preliminaries
We start this section with definitions of properties P, Q, and R for isometries between absolutely continuous function spaces. ese properties are stated for isometries between Lipschitz spaces in [4], page 18, [5], Definition 2, and [2], Definition 2.1.
Suppose that (V, ‖·‖) and (W, ‖·‖) are Banach spaces and X and Y are complete metric spaces. Suppose that T: AC(X, V) ⟶ AC(Y, W) is a linear map. It is said that T has property P, if for every y ∈ Y, there exists v ∈ V such that Tv(y) ≠ 0, where v ∈ AC(X, V) denotes the constant map equal to v. Also, it is said that T has property Q, if for every y ∈ Y and s , there exists v ∈ V such that Tv(y) � w. It is clear that if T has property Q, then it has property P as well.
Definition 1. It is said that the linear map T: AC(X, V) ⟶ AC(Y, W) has property R, if for every y ∈ Y, w ∈ W and ε > 0, there exists v w,ε ∈ V such that ‖v w,ε ‖ ≤ ε and ‖w‖ < ‖w + Tv w,ε (y)‖.
It is easy to see that property R is weaker than property Q and stronger than property P. Also, if T has property P and W is strictly convex, then T has property R.

Results
In this section, we characterize the surjective isometries between vector-valued absolutely continuous function spaces.

Theorem 2. Let X and Y be compact subsets of the real line, and let V and W be Banach spaces with trivial (one-dimensional) centralizers (for definition see [5, 6]). Let T: AC(X, V) ⟶ AC(Y, W) be a surjective linear isometry such that both T and T − 1 have property R. en, T is a weighted composition operator of the form
for all f ∈ AC(X, V) and y ∈ Y, where ϕ: Y ⟶ X and its inverse is absolutely continuous and J is an absolutely continuous map from Y into the space of surjective linear isometries from V into W.
Proof 3. Let C(X, V) denote the set of all continuous maps from X into V, equipped with the supremum norm. By eorem 1, we have for every f ∈ AC(X, V). en, by Lemma 1, we know that every continuous map can be uniformly approximated by absolutely continuous maps. So, we can easily extend T as a surjective linear isometry from C(X, V) into C(Y, W), and we denote this extension again by T.
By [6], Corollary 7.4.11, there exist a homeomorphism ϕ: Y ⟶ X and a map J from Y into the space of surjective linear isometries from V into W such that Tf(y) � J(y)f(ϕ(y)), (6) for all f ∈ C(X, V) and y ∈ Y. Now, consider the unit vector v ∈ V, and take f the constant function equal to v in the equation (6). Since Tf ∈ AC(Y, W), we see that the map y ⟼ J(y)(v) is absolutely continuous and also, the total variation of this map is less than or equal to 1, uniformly with respect to v.
is implies that the map y ⟼ J(y) is absolutely continuous.
On the other hand, it is clear that if f ∈ AC(X, V), then ‖f‖ ∈ AC(X, R).
where v ∈ V is a unit vector and λ > 0 large enough such that x + λ > 0, for all x ∈ X. If we substitute such a function f in the equation (6), we see that ϕ is an absolutely continuous function (note that ‖J(y)(v)‖ � 1). Also, by considering T − 1 , we can show that the inverse of ϕ is absolutely continuous as well. is completes the proof of eorem 2.
It was noticed by the referee that Hosseini [[7], eorem 2.1] has proven a similar result under a weaker assumption on T but under extra assumptions on Banach spaces V and W. Also, it is worth mentioning that during the proof of [7], eorem 2.1, the author uses of [3], eorem 4.1.
Finally, suppose that X and Y are compact subsets of the real line and V and W are strictly convex Banach spaces. Suppose that T: AC(X, V) ⟶ AC(Y, W) is a surjective linear isometry such that T has property P. Since every strictly convex Banach space has trivial centralizer (see [6], eorem 7.4.14) and also by Lemma 3.6 and the proof of Lemma 3.7 in [3] (compare with [4], Remark 4.6), we see that T − 1 has property P as well. en, eorem 2 implies that T is a "weighted composition operator" of the form (6). Notice that if T has property P and W is strictly convex, then T has property R. erefore, by [3], Lemmas 3.14 and 3.15, we are able to recover the main result of [3]. Our proof is somewhat shorter and more elementary than the one in [3].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.