Subspaces containing biorthogonal functionals of bases of different types

The paper is devoted to two particular cases of the following general problem. Let $\alpha$ and $\beta$ be two types of bases in Banach spaces. Let a Banach space $X$ has bases of both types and a subspace $M\subset X^*$ contains the sequence of biorthogonal functionals of some $\alpha$-basis in $X$. Does $M$ contain a sequence of biorthogonal functionals of some $\beta$-basis in $X$? The following particular cases are considered: $(\alpha, \beta)$=(Schauder bases, unconditional bases), $(\alpha, \beta)$=(Nonlinear operational bases, linear operational bases). The paper contains an investigation of some of the spaces constructed by S.Belle\-not in ``The $J$-sum of Banach spaces'', J. Funct. Anal. {\bf 48} (1982), 95--106. (These spaces are used in some examples.)