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2) of the introduction provided that ;3(0) 2': ~ . So the game is to find lower estimates of the Ottaviani constant ;3 (8) defined in the theorem. Note that ;3* (8) is the classical Ottaviani constant and that ;30 (8) 2': ;3* (8) 2': ;3* (8) for all 8 E R . Hence, if Q is subadditive, we see that (B) implies the classical Ottaviani inequality under the independence hypotheses of (B) and that (C) and (D) improve the Ottaviani inequality under the independence hypotheses of (C) and (D). Let t, u E Rand 1::; i ::; n be given.

1. The inclusion S E Ll (H, X) holds if and only if there exist a Hilbert space G, a Hilbert-Schmidt operator T E L(H, G), and an operator R E L(G, X) such that S = RT. Proof. 5), if G is a Hilbert space, then L 1(H,G) = the set of all Hilbert-Schmidt operators from L(H,G). 1) Now sufficiency of the theorem's conditions is evident. Let us prove necessity of the conditions. 2). Let us consider the case of general LCS X. We denote by V the unit ball in H; then the space's ll[H] topology is normalized with the norm 11·11 = oV{-}.

6n ) E En be a given and let us define /-115:= 2:~1 6i/-1i and /-1~:= 2:7=1 6; /-1i . Then we have Sn = S! + S~ and S~ = S! - S~ . Suppose that E and Q(S~)* are independent and that S! and S~ are non-negatively Qi-correlated for all 6 E En and for i = 1,2 . Since Q(± 2x) is B-convex and /-1~ is a B-mean of then by Thm. 5 we have E*Q(2S! + 2/-1~) ::; E*Q(2Sn ) and E*Q( -2S~ - 2/-1t) ::; E*Q( -2Sn ). Since S~ - /-115 = (S! + /-1~) - (S~ + /-1t) and Q is convex, we have s1 , E*Q(S~)::;E* {~Q(2S! + 2/-1~) + ~ Q( -2S~ - 2/-1~)} ::;~ E*Q(2S!

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