Subspaces of LC(H) and C p

Abstract

This paper deals with certain properties of the subspaces of LC(H) and Cp, and namely those connected with the reflexivity and with the property of containing classical spaces. It is proved that any of Cp (1 < p < o) is either isomorphic to Hilbert space or it contains a isomorphic to Ip. For C1 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 Cp (1 < p < oo) be the space of all operators T: H -_ H for which ITIP = trace(T*T)P/2 < o with the norm |. IP LC(H) and Cp are Banach spaces. The word subspace in this paper will mean subspace. For the subspaces of LC(H) and Cp it is known that [1] if X is a 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 X of LC(H) or C1 is isomorphic to H, then it is complemented in L(H), and if X is not isomorphic to H it contains a isomorphic to co or 1 respectively. We will prove results similar to those of Holub for the Cp spaces (1 < p < oo). We shall denote an orthogonal projection from H onto a E by pEE For any two subspaces E, F of H and any 1 < p < oo, REF will denote the of Cp: REF= KA c Cp: (1 PF)A(1PF.) o1. This is closed in the norm | Ip. Proposition 1. If E and F are finite dimensional subspaces of H, then R EF is isomorphic to a Hilbert space for any 1< p < . p Proof~. It is easily seen that REF C REF. 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 46B 10.