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By Robert Schatten

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3. 1. || A // of all . 1. A from t^ into is of finite 7^ "J -norm. III A For an operator (Lemma Theorem operators. 5). 4. By that theorem, II A |L Q = =r sup *Si\&fo sup Y(^^ i8-) ^>5 f The extreme right clearly represents the bound of A H M I 8 I This concludes the . proof. 2. operators ( f may 7^ Cfyy )* rom 1^ into 7i , where be interpreted as the Banach the norm of an operator equals to its bound. Proof. 3. ' A 5. "natural equivalence*. REMARK fy for ( f <8 g which CROSS-SPACES OF OPERATORS III.

Clearly, an expression ^> an operator A of finite ^ g^ in f. 8, also every operator A of finite rank. of finite rank many such expressions. To mined by where assumed ajF) = may sum space this if ^ for F In the case when *^j from T^Jinto 72 is e , ...... , We g,^. ". elements Thus, f , ....... f A is of the then determined by A A(F) ^y f^ in Tji is = deter- S^ Since "^ 1^. determined by considered as a subspace they fur- if, is reflexive,, have, up as follows: For any two Banach spaces ^"f^may be .

Fc, G |f ...... G ^ while , /v fif G ' * are in ). these expressions, in particular the invariance of their "rank" under equivalence, we g^ The proof follows from the preceding discussion. Similarly, (where 3Efj. may be found in l2j . In the present chapter only state those properties of expressions which will be needed in our future discussions. Transformations on expressions. 3. 2. *. # ZT^ vSf i* Proof. For a given expression JS'IT ^L^ 8, ^gJ the same (i)--(iii). = JE"J^, F(fjg c Since . for both sides of any of the This means F .

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A theory of cross-spaces by Robert Schatten


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