Extreme points and isometries on vector-valued Lipschitz spaces

Isometries of Lipschitz type function spaces

“…In fact, we use the concept of property in order to show that T preserves the supremum norm in the second section. Then, we extend the main results of and , simultaneously in one proof, in a more general setting. Moreover, this provides an affirmative answer to a question raised by the author in [12, Question 1.9] (see the fifth section).…”

“…First, we recall the definitions of and properties (see and ). Suppose that and are Banach spaces, X and Y are complete metric spaces.…”

“… Abstract In this article, we describe isometries over the Lipschitz spaces under certain conditions. Indeed, we provide a unified proof for the main results of and in a more general setting. Finally, we extend our results for some other functions spaces like the space of vector‐valued little Lipschitz maps and pointwise Lipschitz maps.…”

“…Suppose that X and Y are compact metric spaces and V and W are quasi sub‐reflexive Banach spaces with trivial centralizers. 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.…”

“…Suppose that and are compact metric spaces and and are quasi sub-reflexive Banach spaces with trivial centralizers. In [5], it is shown that every surjective linear isometry ∶ ( , ) → ( , ) which both and −1 have property  (for definition, see the second section), is a "weighted composition operator" of the form…”