Surjective isometries on vector-valued differentiable function spaces

Isometries on Banach algebras of vector-valued maps

“…As U 0 is a surjecton and B 2 separates the points of X 2 × Y 2 , there exists a function F ∈ B 1 such that U 0 (F )(z, y) = U 0 (F )(w, y). Then by (24) we have…”

“…Jiménez-Vargas and Villegas-Vallecillos in [17] have considered isometries of spaces of vector-valued Lipschitz maps on a compact metric space taking values in a strictly convex Banach space, equipped with the norm f = max{ f ∞ , L(f )}, see also [16]. Botelho and Jamison [3] studied isometries on C 1 ([0, 1], E) with max x∈[0, 1] [32,26,18,1,2,23,6,31,5,27,19,20,21,24,22,25,15] From now on, and unless otherwise mentioned, α will be a real scalar in (0, 1). Jarosz and Pathak [14] studied a problem when an isometry on a space of continuous functions is a weighted composition operator.…”

Isometries between Spaces of Vector-Valued Differentiable Functions

“…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.…”

“…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 differentia-ble 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 f∂ γ TgðτðxÞÞ: γ ∈ Γg must have at most one nonzero term?…”

“…Also, Li and Wang [7] studied the isometries between spaces of p-times differentia-ble 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 f∂ γ TgðτðxÞÞ: γ ∈ Γg must have at most one nonzero term? Compare with the proofs in ([4], Lemma 2.5) and ([2], Lemma 2.3).…”