A class of regular matrices

Author:
Godfrey L. Isaacs

Journal:
Proc. Amer. Math. Soc. **60** (1976), 211-214

MSC:
Primary 40C05

DOI:
https://doi.org/10.1090/S0002-9939-1976-0430589-4

MathSciNet review:
0430589

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *m* be the space of real, bounded sequences $x = \{ {x_k}\}$ with the sup norm, and let $A = ({a_{n,k}})$ be a regular (i.e., Toeplitz) matrix. We consider the following two possible conditions for *A*: (1) $\Sigma _{k = 1}^\infty |{a_{n,k}}| \to 1$ as $n \to \infty$, (2) $\Sigma _{k = 1}^\infty |{a_{n,k}} - {a_{n,k + 1}}| \to 0$ as $n \to \infty$. G. Das [J. London Math. Soc. (2) **7** (1974), 501-507] proved that if a regular matrix *A* satisfies both (1) and (2) then (3) ${\overline {\lim } _{n \to \infty }}{(Ax)_n} \leqslant q(x)$ for all $x \in m$, where $q(x) = {\inf _{{n_i},p}}{\overline {\lim } _{k \to \infty }}{p^{ - 1}}\Sigma _{i = 1}^p{x_{{n_i} + k}}$. Das used “Banach limits” and Hahn-Banach techniques, and stated that he thought it would be “difficult to establish the result... by direct method". In the present paper an elementary proof of the result is given, and it is shown also that the converse holds, i.e., for a regular *A*, (3) implies (1) and (2). Hence (3) completely characterizes the class of regular matrices satisfying (1) and (2).

- Richard G. Cooke,
*Infinite matrices and sequence spaces*, Dover Publications, Inc., New York, 1965. MR**0193472** - G. Das,
*Banach and other limits*, J. London Math. Soc. (2)**7**(1974), 501–507. MR**336148**, DOI https://doi.org/10.1112/jlms/s2-7.3.501 - G. L. Isaacs,
*An iteration formula for fractional differences*, Proc. London Math. Soc. (3)**13**(1963), 430–460. MR**155121**, DOI https://doi.org/10.1112/plms/s3-13.1.430 - G. G. Lorentz,
*A contribution to the theory of divergent sequences*, Acta Math.**80**(1948), 167–190. MR**27868**, DOI https://doi.org/10.1007/BF02393648

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
40C05

Retrieve articles in all journals with MSC: 40C05

Additional Information

Keywords:
Regular matrices,
Toeplitz matrices,
Banach limits,
almost convergence

Article copyright:
© Copyright 1976
American Mathematical Society