SUBSPACES OF LCiH) AND Cp

This paper deals with certain properties of the subspaces of LC(H) and C and namely those connected with the reflexivity and with the property of containing classical spaces. It is proved that any subspace of C (1 < p < oo) is either isomorphic to Hilbert space or it contains a subspace isomorphic to I . For Cj and LC'H) the same results were obtained by J. R. Holub, cf. [4]. Introduction. Let H be a complex Hilbert space. Denote by LC(H) or C the space of all compact operators on H, with the usual operator norm. Let C (l < p < oo) be the space of all operators T: H —> H for which \T\P = traceiT*T)p < oo with the norm | • | . LCiH) and C are Banach spaces. The word "subspace" in this paper will mean ' closed subspace". For the subspaces of LC(H) and C it is known that [l] if X is a subspace of LC(H) which is complemented in the space L(H) of all bounded operators on H, then X is a reflexive space. J. R. Holub [4] proved that if a subspace X of LC(H) or Cj is isomorphic to H, then it is complemented in L(H), and if X is not isomorphic to H it contains a subspace isomorphic to cn or I. respectively. We will prove results similar to those of Holub for the C spaces (l < p < oo). We shall denote an orthogonal projection from H onto a subspace E by PEp p For any two subspaces E, F of H and any 1 < p < oo, R will denote the subspace of C : «fF = {A e C ■ (1 PJAQ PP) = Of. P pth This subspace is closed in the norm I • I . Proposition 1. // E and F are finite dimensional subspaces of H, then E F R is isomorphic to a Hilbert space for any 1 < p < °°. Proof. It is easily seen that RM CRj . Presented to the Society, July 30, 1973; received by the editors July 22, 1974 and, in revised form, September 18, 1974. AMS (MOS) subject classifications (1970). Primary 47B05, 47B10; Secondary 46B10.