Geodesic
Approximation of Lp -Contractions by Isometries
Canadian mathematical bulletin, Tome 32 (1989) no. 3, pp. 360-364

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We 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 L 1-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 
Akcoglu, M. A.; Boivin, D. Approximation of Lp -Contractions by Isometries. Canadian mathematical bulletin, Tome 32 (1989) no. 3, pp. 360-364. doi: 10.4153/CMB-1989-052-1
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     title = {Approximation of {Lp} {-Contractions} by {Isometries}},
     journal = {Canadian mathematical bulletin},
     pages = {360--364},
     year = {1989},
     volume = {32},
     number = {3},
     doi = {10.4153/CMB-1989-052-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1989-052-1/}
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