2-Local standard isometries on vector-valued Lipschitz function spaces

Abstract

Under the right conditions on a compact metric space X and on a Banach space E, we give a description of the 2-local (standard) isometries on the Banach space Lip(X, E) of vector-valued Lipschitz functions from X to E in terms of a generalized composition operator, and we study when every 2-local (standard) isometry on Lip(X, E) is both linear and surjective.

2-local isometries on function spaces

“…After the publication of [10] some authors expressed their suspicion about the argument there and the validity of the statement there had not been confirmed until the correction [8,Corollary 15] was published by Hatori and Oi. In this section by applying [8,Lemmas 10,11] and [5,Proposition 7] we exhibit the form of a surjective real-linear isometry between the Banach algebras of Lipschitz functions.…”

Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital Commutative C⁎-Algebras

“…Botelho and Jamison [10] studied isometries on 1 ([0, 1], ) with max ∈[0,1] {‖ ( )‖ + ‖ ( )‖ }. See also [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Refer also to a book of Weaver [28].…”

Isometries on Banach algebras of vector-valued maps

“…Then by (12) and (13) That is ψ θ is the point evaluation for I( B 2 ) at (x c , y c , m c , γ c ). By (17), (19), (15) and (16) 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.…”