Reflexivity of sets of isometries on bounded variation function spaces

Abstract

For arbitrary subsets X and Y of the real line with at least two points, let BV(X) (resp. BV(Y)) be the Banach space of all functions of bounded variation on X (resp. Y) endowed with the natural norm $\Vert \cdot {\Vert }_{\mathrm{\infty }}+\mathcal{V}(\cdot )$, where $\Vert \cdot {\Vert }_{\mathrm{\infty }}$ and $\mathcal{V}(\cdot )$ denote the supremum norm and the total variation of a function, respectively. We show that the set of all surjective linear isometries from BV(X) onto BV(Y) is topologically reflexive. Among the consequences, it is also proved that the set of all isometric reflections, and the set of all generalized bi-circular projections on BV(X) are topologically reflexive.

Keywords:

• Surjective linear isometry
• generalized bi-circular projection
• functions of bounded variation
• algebraic reflexivity
• topological reflexivity

2010 Mathematics Subject Classifications:

• Primary 47B38
• Secondary 46J10
• 47B33

Acknowledgments

The author would like to thank the referees for their invaluable comments which definitely helped to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).