ON WEIGHTED COMPOSITION OPERATORS BETWEEN SPACES OF MEASURABLE FUNCTIONS

Abstract

In this note we provide an elementary proof of the following: every automorphism S of L ^∞ is of the form Sf = f o Sz for all f ε L ^∞. As a first corollary we prove that if there exists a linear isometry T of a linear subspace A of L ^∞ containing H ^∞ onto such a subspace B, then T can be written as a weighted composition map, namely, Tf = α (f o Sz) for all f ε A, where α ε B, |α(λ)| = 1 for all λ in the unit circle and S is an automorphism of L ^∞ induced by T. As a straightforward consequence, we obtain a description of the linear isometries between Douglas algebras. As a second corollary we show that every linear bijection T of L ^∞ onto L ^∞ which preserves non-vanishing functions can also be written as a weighted composition map.