A-associate spaces and matrix transformations (Q1067580)

Let X be a sequence space and A a fixed infinite matrix. The A-associate space of X is the set of all \(y=\{y_ k\}\) such that for each y, \(\lim_{n\to \infty}\sum^{\infty}_{k=1}a_{nk} x_ k y_ k\) converges whenever \(x=\{x_ k\}\in X\). In section 1, some examples of A- associate spaces will be given. We shall also consider the AL property of a sequence space which was suggested by Zeller. Then we shall show that a sequence space having the AK property also has the AL property under certain conditions. In section 2, we shall give some consequences of a sequence space X having the AL property. By rephrasing Theorem 4 in [\textit{Lee Peng Yee}, ibid., Spec. Issue, 102-108 (1979; Zbl 0428.46011)] in terms of AL property, we shall prove the main theorem. In section 3, we shall consider the matrix transformations from a BK-space into spaces \((\ell_{\infty})_ B\), \(c_ B\) and \((\ell_ 1)_ B\), where B is an infinite lower semi-matrix with a nonzero diagonal.