A theory of cross-spaces - download pdf or read online

By Robert Schatten

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Ft g. it. equivalent to = Tf/C and . equivalent to is also where Pf . the closed linear T^

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 . For an operator does not exist, we define justification of such a definition will be clear || A {/ .

A ing" operator from Ill For a fixed additive and bounded functional on A fromt^into 4( That || f g = ) = Ill^lil 3U|^ in the proof of Theorem is = ipj same fy shown oC-norm finite = III 11^ 'J[ , ^y ( ( $fll f g f g This can be . 1. in addition to ) ^/ft (g) For "corresponding* operators the A U Thus, . 1, Furthermore, . ^j (A)g , from The same y . 2 and By . { oC-norm of finite = If X we ]f^ (that is, generated by also have 111 A = Iff )/| A |/| . This can be seen as follows: III A = III sup II Af || = sup 14 sup Hfli*l ( f g )| = 1*11*11311*1 ||Xg|| = III A .

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