The kernel of a sequence obtained from a given sequence with the aid of a regular transformation

The smallest set is found that contains the kernel of a sequence obtained from a sequence of elements $\{x_n\}$ of a Banach space with the aid of a regular transformation of the class $T(C,C')$. Here $T(C,C')$ is the set of complex matrices $(c_{nk}\equiv(a_{nk}+ib_{nk})$ satisfying the conditions $\varlimsup\limits_{n\to\infty}\sum_{k=1}^\infty|a_{nk}|=C\ge1$, $\varlimsup\limits_{n\to\infty}\sum_{k=1}^\infty|b_{nk}|=C'\ge0$.