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It was proved by R. Wittmann [2] that, given a positive linear contraction of LP (1 ap or 0; the (best possible) value of ap is the lp-norm of a certain 3 x 3 matrix. In this paper ap is explicitly expressed as a function of p . Let T be a positive linear contraction of LI . Ornstein and Sucheston [1] proved that suplif I ap or 0. Here the constants ap are defined by ap= sup ((1x)P(1+ )1 (I+2P-')IIP (b) ap = aq, (1 < p, q < o0, 1/p + 1/q = 1), (C) a2 = A3. In this paper we determine the exact value of ap; namely, we prove the following: Theorem. The constants ap are given by (1) _ f (1 + 2P-I)l/P 1 <,p < 2, ap (1 + 211(P-1))1 -/P 2 <p < oc. Proof. In view of the second definition of ap we have ap = max{Ix YIP + ly ZIP + IZ xIP: IXIP + IYIP + IZIP = 1}. Received by the editors June 3, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E30, 47B38.

A note on the 𝐿^{𝑝} analogue of the “zero-two” law