Contractive Projections in Nonatomic Function Spaces

Abstract

Abstract. We prove that there is no 1-complemented subspaces of finite codimension in separable rearrangament-invariant function spaces.We study contractive projections onto finite-codimensional subspaces of real nonatomic function spaces. In general such projections are not common. It is well known that only In this paper we prove that in rearrangement-invariant nonatomic function spaces not isometric to L 2 there is no 1-complemented subspaces of any finite codimension.We use the terminology and notation as in [9].Our method of proof is surprisingly simple -it is based on the following observation: Proposition 1. (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. However the original theorem uses special technical frazeology so we state it below in the language of projections:1991 Mathematics Subject Classification. 46B,46E.

Norm-One Projections in Banach Spaces

“…[Ru91]) and facts related to Proposition 5.16. Moreover, in [KaRa94,Ra95] the following fact is used: Proposition 5.41. (cf.…”

On the essential norms of Toeplitz operators with continuous symbols

“…and it is known that I − K 0 X→X > 1 for every separable rearrangementinvariant Banach function space X not isometric to L 2 (T) (see [31,Theorem 4]). However, the latter does not immediately imply that T (e −1 ) H[X]→H[X] > 1, since the inequality I − K 0 H[X]→H[X] ≤ I − K 0 X→X is, in general, strict.…”

Convex-transitive characterizations of Hilbert spaces

“…Hence Randrianantoanina's question (Q R ) from the introduction involves the existence of Flinn elements in the convex-transitive setting. See [14] for connections between Flinn pairs and numerically positive operators and [13] for related results.…”