On Complementability of Subspaces in Symmetric Spaces with the Kruglov Property

We show that, for a broad class of symmetric spaces on $[0,1]$, the complementability of the subspace generated by independent functions $f_k$ $(k=1,2,\dots)$ 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 $Z_X^2$ on the semiaxis $[0,\infty)$. Moreover, if $\sum_{k=1}^\infty m(\operatorname{supp}f_k)\le 1$, then $Z_X^2$ 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 $L_p[0,1]$ $(1\le p<\infty)$ of the subspace $[f_k]$ generated by independent functions provided that it is isomorphic to the space $l_p$ is obtained.