Surjective Isometries on Rearrangement-Invariant Spaces

Abstract

Abstract. We prove that if X is a real rearrangement-invariant function space on [0, 1], which is not isometrically isomorphic to L 2 , then every surjective isometry T : X → X is of the form T f (s) = a(s)f (σ(s)) for a Borel function a and an invertible Borel mapIf X is not equal to L p , up to renorming, for some 1 ≤ p ≤ ∞ then in addition |a| = 1 a.e. and σ must be measure-preserving.

Norm-One Projections in Banach Spaces

“…The main tool in proofs of Theorem 7.38, 7.39 and 7.40 is Proposition 5.41, which allows to transfer to the real space setting techniques analogous to hermitian elements in complex spaces (see [KaRa94]). …”

A note on the Banach-Mazur problem

“…[12,19]) and, as mentioned above, they are almost transitive. Theorem 1.1 generalizes a recent result of Skorik and Zaidenberg:…”

Contractive Projections in Nonatomic Function Spaces

“…(cf. [7,11]) In a real Banach space X if P is a projection then I − P = 1 (where I denotes identity operator) if and only if x * (P x) ≥ 0 for all x ∈ X and x * ∈ X * norming for x.In [7, Theorem 4.3] (cf. [8,10]) Kalton and the author proved the nonexistence of 1-complemented hyperplanes in a wide class of nonatomic function spaces.…”

“…In [7,Theorem 4.3] (cf. [8,10]) Kalton and the author proved the nonexistence of 1-complemented hyperplanes in a wide class of nonatomic function spaces.…”

“…[8,10]) Kalton and the author proved the nonexistence of 1-complemented hyperplanes in a wide class of nonatomic function spaces. However the original theorem uses special technical frazeology so we state it below in the language of projections:…”