On complementability of subspaces in symmetric spaces with the Kruglov property

Abstract

We show that, for a broad class of symmetric spaces on [0, 1], the complementability of the subspace generated by independent functions f _k (k = 1, 2,…) is equivalent to the complementability of the subspace generated by the disjoint translates $\bar f_k (t) = f_k (t - k + 1)\chi _{[k - 1,k]} (t)$ of these functions in some symmetric space Z ² _X on the semiaxis [0,∞). Moreover, if Σ ^∞ _k=1 m(supp f _k ) ⩽ 1, then Z ² _X can be replaced by X itself. This result is new even in the case of L _p -spaces. A series of consequences is obtained; in particular, for the class of symmetric spaces, a result similar to a well-known theorem of Dor and Starbird on the complementability in L _p [0, 1] (1 ⩽ p < ∞) of the subspace [f _k ] generated by independent functions provided that it is isomorphic to the space l _p is obtained.