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Almost-coercive matrix transformations. (English) Zbl 0732.40002
Following C. Eizen and G. Laush [Math. Jap. 14, 137-143 (1969; Zbl 0195.067)], a sequence-to-sequence matrix transformation A is said to be almost-coercive if the A transform of every bounded sequence is almost-convergent in the sense of G. G. Lorentz. By reference to the characterization that was given by J. P. Duran [Math. Z. 128, 75-83 (1972; Zbl 0235.40004)] for this type of matrices, the authors formalize corresponding conditions on series-to-sequence and series-to-series transformation matrices. (Yet, when detailing the result of Eizen and Laush, some condition - being claimed by Duran to be inadequate and thus rendering their theorem false - presumably is taken for an equivalent to Duran’s stronger one.) Theorem 2.4, while worked in the very terms of the Corollary to Theorem 1 (not given reference to, though), in Duran’s paper, is in fact its series-to-sequence version.
40C05Matrix methods in summability