Kernels of bounded sequences

Necessary and sufficient conditions are obtained for a regular positive matrix to leave unchanged the kernel of a given bounded sequence. Let $\|a_{nk}\|$ be a positive $T$-matrix, let $\{S_n\}$ be a bounded sequence of real numbers, and let $\tau_n=\sum_{k=0}^\infty a_{nk}S_k$. In order that $\underline{S}=\varliminf\limits_{n\to\infty}S_n=\varliminf\limits_{n\to\infty}\tau_n(\overline{S}=\varlimsup\limits_{n\to\infty}S_n=\varlimsup\limits_{n\to\infty}\tau_n)$, it is necessary and sufficient that, for any $\varepsilon>0$, there exist sequences $\{m_k\}$ and $\{\nu_j\}$ such that $|S_{\nu_i}-\underline{S}|\le\varepsilon$ ($|S_{\nu_i}-\overline{S}|\le\varepsilon$) $(i=1,2,\dots)$ и $\sum_{i=1}^\infty a_{m_k\nu_i}\to1$ $(k\to\infty)$.