Matrix maps of quasi-convex null sequences

ISSN： 0236-5294

Source： Acta Mathematica Hungarica, Vol.94, Iss.3, 2002-04, pp. : 197-210

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Abstract

In [7] Stieglitz and Tietz identify the space q ^α of all quasi-convex convergent sequences as a BK-space. They characterize all infinite matrices which map q ^α into an arbitrary FK-space. In [6] they do so for matrices which map a particular class of sequence spaces into q ^α. In [10] Zygmund introduces q ² in connexion with convergence factors of Fourier series. Dawson considers in [3] and [4] matrix maps of the space q ₀^α of all quasi-convex null sequences. In Section 2 we characterize all matrices which map q ₀^α into an arbitrary FK-space. Prior to that, a particular matrix map on q ₀^α gives us the BK-topology on q ₀^α . As an application we characterize in Section 3 the matrices which map q ₀^α into the FK-spaces considered by Stieglitz and Tietz in [8]. Based on [6], we determine the matrices which map these spaces into q ₀^α . Using methods similar to those in [7] our results in Section 2 depend on Theorems 2.1 and 4.1 in [5] due to Jakimovski and Livne. Theorem 2.1 gives for suitable pairs of sequence spaces necessary and sufficient conditions for an infinite matrix to map one space into the second one. In Theorem 4.1 a special sequence which is useful in applications of quasi-convexity is constructed. We close our paper with two remarks concerning three results in [8].