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

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) over bar (k)(t) = f(k)(t - k + 1)chi([k-1, k))(t). Moreover, if Sigma(infinity)(k=1) m(supp f(k)) <= 1, then Z(X)(2) can be replaced by X itself in the last statement. This result is new even for L-p-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 < infinity) of the closed linear span of some independent functions under the assumption that this closed linear span is isomorphic to l(p).