ON SUBSPACES GENERATED BY INDEPENDENT FUNCTIONS IN SYMMETRIC SPACES WITH THE KRUGLOV PROPERTY

摘要

For a broad class of symmetric spaces X, it is shown that the subspace generated by independent functions f_k (k = 1, 2,...) is complemented in X if and only if so is the subspace in a certain symmetric space Z_X~2 on the semiaxis generated by their disjoint shifts f_k(t) = f_k(t - k + 1)χ_([k-1,k))(t). Moreover, if ∑_(k=1)~∞ m(suppf_k) ≤ 1, then Z_X~2 can be replaced by X itself in the last statement. This result is new even for Lp-spaces. Some consequences are deduced; in particular, it is shown that symmetric spaces enjoy an analog of the well-known Dor-Starbird theorem on the complementability in L_p[0, 1] (1 ≤ p < ∞) of the closed linear span of some independent functions under the assumption that this closed linear span is isomorphic to ?_p.