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By Robert Schatten

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F A II Let9* a implies ]j A^ ^ aA implies whenever 3 ^r - H* 9^ ^r? *Jj ^< ^^ ) *** and 9*111 II' denote its bound. > ( 1 ^ = &(t ) \\ _ Ci / ) t * *j (^, g) Clearly, k6 CROSS-SPACES OF OPERATORS 111. ll ) II g unique element of 6i J^, a f III^H! ( for II g 1^ which we ) for g f ^,, g (ii) is additive. Since 7^ is the Af ( ) That functional . ^ . 2 however, that least constant denoted by |1 A bounded since 11^ 111 A Therefore, . 5, above, \\\<$ for which A f( A from of (a) above. is ( \\\ A <+oo.

Of the greatest It is . 1. , p. 138, for of termed a is complementary manifold. 2. projection in a same time an ' not smaller than the value for that T*> ( O ^^ . 54 CROSS-SPACES OF OPERATORS III. 6. ** such that T/= Ttt, 7u is the conjugate space of Suppose further . ) denote a Banach space and *WT a closed linear "Y^ that, TTU,. C T^'W, crossnorm on ^^uLc without changing which (fg) = =|||il|| = ^r ffcHT, 8 eTTLo , and .

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.

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A theory of cross-spaces by Robert Schatten


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