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.
Abstract. This is the survey of results about norm one projections and 1-complemented subspaces in Köthe function spaces and Banach sequence spaces. The historical development of the theory is presented from the 1930's to the newest ideas. Proofs of the main results are outlined. Open problems are also discussed. Every effort has been made to include as complete a bibliography as possible.
We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of c 0 -, 1 -and ∞ -sums and giving as a new result the case of E-sums where E has the RNP and n(E) = 1 (in particular for finite-dimensional E with n(E) = 1). We also show that the numerical index of a Banach space Z which contains a dense union of increasing one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that, for a fixed p, the numerical indices of all infinite-dimensional L p (μ)-spaces coincide.
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