A note on isometries of Lipschitz spaces

Isometries of Lipschitz type function spaces

“…In , it is shown that every surjective linear isometry which both T and have property (for definition, see the second section), is a “weighted composition operator” of the form where is a bi‐Lipschitz homeomorphism and J is a Lipschitz map from Y into the space of surjective linear isometries from V into W . In , by removing the quasi sub‐reflexivity assumption from the main result of Botelho, Fleming and Jamison [5, Theorem 6], the author extends this result.…”

“…where ∶ → is a bi-Lipschitz homeomorphism and is a Lipschitz map from into the space of surjective linear isometries from into . In [12], by removing the quasi sub-reflexivity assumption from the main result of Botelho, Fleming and Jamison [5,Theorem 6], the author extends this result.…”

“…Note that every Lipschitz map which is constant outside a given ball, belongs to the little Lipschitz space (using the condition ). Also, since metric spaces are compact, in the process of the proof of Theorem , we may avoid to use Lemma (see the proof of Proposition 1.3 in ). Similar to the proof of Theorem , by using the function , for , when , we can show that (note that ).…”

“…Note that every Lipschitz map which is constant outside a given ball, belongs to the little Lipschitz space (using the condition < 1). Also, since metric spaces are compact, in the process of the proof of Theorem 2.5, we may avoid to use Lemma 2.2 (see the proof of Proposition 1.3 in [12]).…”

Isometries on Banach algebras of vector-valued maps

“…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.…”