On the exactness of a theorem of Mazur and Orlicz

N. A. Davydov

Matematičeskie zametki, Tome 11 (1972) no. 4, pp. 431-436

Résumé

For a preassigned unbounded sequence $\{S_n\}$ of complex numbers, and preassigned complex numbers $z_1$ and $z_2\ne z_1$ we consruct: 1) regular matrices $A=||a_{nk}||$ and $B=||b_{nk}||$ such that the same bounded sequences are summable by these matrices and that $\lim\limits_{n\to\infty}S_n=z_1(A)$, $\lim\limits_{n\to\infty}S_n=z_2(B)$; 2) regular matrices $A^{(1)}=||a^{(1)}_{nk}||$ and $B^{(1)}=||b^{(1)}_{nk}||$ such that $B^{(1)}\subseteq A^{(1)}$, $\lim\limits_{n\to\infty}S_n=z_1(A^{(1)})$ and $\lim\limits_{n\to\infty}S_n=z_2(B^{(1)})$. Our results show that the well known theorem of Mazur–Orlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.