Approximation of Lp-Contractions by Isometries

Abstract

AbstractWe construct a positive linear contraction T of all LP(X, μ)- spaces, 1 ≦ p ≦ ∞, μ(X) = 1 such that T1 = 1, T* 1 = 1 and also Tf > 0 a.e. for all f ≧ 0 a.e., f ≢ 0 but for which there is an f ∊ L∞ such that (Tnf — ∫ fdμ) does not converge in L1-norm. We also show that if T is a contraction of a Hilbert space H, there exists an isometry Q and a contraction R such that ∥Tnx - QnRx∥ —> 0 as n —» ∞ for all x in H