# A theory of cross-spaces by Robert Schatten

By Robert Schatten

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TflS*U| llFll OT*(G)H|||T*|I| ||G|| = IIISHI = III Let S Their adjoints S satisfies condition VI. denote two operators on ^, and Irrespectively. |fsV)/l " 8 " and T T Clearly, and llFll llGll Til/ and . sf. Tgj^iilsHi F ii/Tiii , . ^Bi) This concludes the proof. LEMMA Proof. 7. Let F A is a and G 4MM that for any expression^-C^fj crossnorm. be fixed. 4 52 CROSSNORMS II. 5 LEMMA F *G *X ( ^ ) G F II = ) II G If F (J G II II G- . tf This together . 1, ^ is a This concludes the proof. 1. 8. F. 9*4 Lemma crossnorm by >'( proves that is finite for A $ IIFII llGfl with a crossnorm oc' ^^ O6 is also a crossnorm.

Afc)c The corre- K-8 Cy) is 50 CROSS-SPACKS OF OPERATORS III. ) llAll,,. This concludes the proof. 4. be interpreted as the Banach space of oO-norm of finite (with be approximated in that A Proof. ^ Let ot be a given crossnorm )) A 11^ norm all "X norm Then, A operators representing the . ^x from A ), Tr^ into T^ which may- by operators of finite rank. , norms to the limit of the senting quence it. 2, the given element. norm of such an element equals expressions in the fundamental sequence repre II A IL~ AP A A intoTJ^for which .

4. 6. 8/^ into 2Ci F fJ G(g^ ( is *)^ = or 2T^,F(fJg^ g ~ *^ = d satisfies condition That define numbers obtained when all J= F the expression 2E^ f f c a uniform crossnorm. "^ for all as represents the bound of the operator l^determined by Proof. 4 gives 2l7I|f we 1 ^ if, T? varies in T ^, and Irrespectively. 8. Thus, I. satisfies conditions if, if immediate. We and if, 9^ CROSSNORMS II. 4. shall check condition V: >(f = g) (8)1 -UPJJ JJ G|[ F(f)| I " ffl Finally we shall prove that ^ represent operators onl^, and Irrespectively.