By Robert Schatten

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9. 1. 10 the last UP *>^ Hfll Hg | number equals to >(3EjI,^ II This concludes &f) the proof. THEOREM also a crossnorm. Then, oc" Proof. 1. 2. ' csC , * ...... 11 Thus, by Theorem norms. This concludes the proof. is are also crossnorms. are crossnorms, Applying successively Od/'^>on Tg*OT&* ..... 1, get OC/ <** . ^ A by ^>^on <*? ~~ Tfi , are cross- CROSSNORMS II. 1 states that associate is also a crossnorm. sent the least crossnorm. when in 1J. 35 We remark In fact, we that "X does not necessarily repre- shall prove later (Chapter V, % 11) that and 1-^denote two two-dimensional Euclidean spaces a least crossnorm 1^01^^ does By Theorem not exist. ~~

5 gives lence. II Zl7*F(fJg Now suppose F pose that = II is ^ ^(Zr= F|l g 1 . gj is a and such that Similarly we can find FC^with equivalent to Z^^tfjg^ > consequence 2^, h c k^ & <- Z^ f c * (2L^i f i . gj F)| By Lemma Naturally 8c = 1 . 5 for ^ . ) t Consequently, immediate and IV we can , is not . C "^*"for Let 2T7^, f c Ill: Clearly, || that JSE-jL, f^ there exists an l| - %> >Q 58 CROSSNORMS II. 4 give, The ^ last inequality holds for any V: - F(f) Let llf :, and II f gc c llF C^ = II f 1 g We .

The relation a projection of 1^* on I^Cwith bound manifold of f^ This concludes the proof. 7. i Then, the existence of a projection of for any an extension of Thus the bound (6). iii (2), (4) LEMMA P prove that (5) Furthermore, . pin by virtue of since and (1), (3) 55 ST^fT f/fe"fC * g . To for Thus, and therefore, The left side clearly, represents an expression in side is an expression in TfcdC "() OC , while the right Thus, both sides must be equivalent to ft CROSS-SPACES OF OPERATORS III. ^ fcl v Since Thus, the *ft@OC P inf - o ^_, i, equivalent to^ET^fTlS g?