By Robert Schatten
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Extra info for A theory of cross-spaces
We dimensional. 5 and 3. 2. 3). Chapter A operator g^. in and term the Ij^ which such a The (not necessarily where the norm ), when we deal with o(/-norm. 06 stand for a crossnorm on let from 7^ there exists a finite constant if expressions^T^ fj& denote by Theorem of this or Tj^is although this is not stated each time explicitly. 7^^ for all 1^>. Q Throughout the rest *pj into T^, This bound. ^ associate space. ^* O into of finite such that The least o(/-norm* Ti*is termed of A of such constants .
Proof. I: Let2L* ( with F II l\ By Lemma = we 1 IJi* . 2 the F For an denote a fixed expression. 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.
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.