Limits of indeterminacy of sequences obtained from a given sequence using a regular transformation

The problem considered is how there can be a set of weak accumulation points of the subsequences of a sequence obtained from a given sequence by using a regular transformation of the class $T(C,C')$ when the terms of the sequences are elements of a reflexive Banach space. $T(C,C')$ denotes the class of complex regular matrices $c_{mn}$ ($c_{mn}=a_{mn}+ib_{mn}$, where $a_{mn}$ and $a_{mn}$ are real numbers) satisfying the conditions $\varlimsup\limits_{m\to\infty}\sum_{n=0}^\infty|a_{mn}|=C$ и $\varlimsup\limits_{m\to\infty}\sum_{n=0}^\infty|b_{mn}|=C'$