tailieunhanh - Báo cáo toán học: "A result on operators on $\cal C$ [0,1] "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một kết quả trên các nhà khai thác trên $ \ cal C $ [0,1]. | J. OPERATOR THEORY 3 1980 275-289 Copyright by INCREST 1980 A RESULT ON OPERATORS ON ê 0 1 J. BOURGAIN INTRODUCTION Throughout the text subspace means always infinite dimensional subspace . For generalities about Banach spaces we refer to 9 . m stands for 0 1 . Let r denote the Rademacker functions on 0 1 . We recall that a Banach space X has cotype q 2 q oo iff there exists a constant p 0 such that for all finite sequences x15 . x of elements of It is known that X has cotype q for some q oo if and only if X does not contain n 1 2 . uniformly or in other words c0 is not finite dimensionally representable in X see 10 for instance . The following result is due to H. p. Rosenthal 4 or 14 . Theorem 1. Let 9 be a Banach space and T .m an operator such that T ls not separable Then there exists a subspace X isometric to cể such that T x is an isomorphism. If X is a subspace of and XÍ a subset of T we say that xe is norming for X provided sup I X àp X x for all xe X . nèx J j identifying with the Radon measures p on 0 1 . We will prove Theorem 2. If X is a cotype subspace ofcể andxf a w -compact subset of ể which norms X then Xf is not separable. Taking Xf T y y IIjAI 4 with M big enough we obtain as immediate consequence of Theorem 1 and Theorem 2 276 J. BOURGA1N Corollary 3. If is a Banach space and T J6 y an operator fixing a cotype subspace of then Tfixes a copy of d. Applying 14 Corollary 1 we get also Corollary 4. Any complemented subspace of Ý which has a cotype subspace is isomorphic to 6. One may conjecture that Theorem 2 also holds under the weaker hypothesis that c0 does not imbed in x. The rest of the paper is devoted to the proof of Theorem 2. Since Ỹ and if d the continuous functions on the Cantor set are isomorphic see 11 we may replace 6 by tf d . REDUCTION TO THE CASE OF POSITIVE MEASURES Denote by JI d the space of Radon measures on A 0 1 N. If p e . d then Jpi is the variation of p. It is clear that for a norm-separable w -compact subset Jf of