Multiplication on the generalized analytic sequences and matrix transformations (Q1194637)

Let \(p=(p_ n)\) be a sequence of strictly positive real numbers and \(\nu=(\nu_ n)\) any fixed sequence of non-zero complex numbers such that the sequence \((|\nu_ n|)\) is nondecreasing and \(\liminf_ n|\nu_ n|^{-p_ n}=r\in[0,\infty)\). The following sequence spaces are introduced: \(D_ \infty^ \Lambda(p)=\{(x_ n)\): \(|\nu_ n x_ n|^{p_ n}=O(1)\}\), \(D_ 0^ \Lambda(p)=\{(x_ n)\): \(|\nu_ n x_ n|^{p_ n}=o(1)\}\). The authors derive necessary and sufficient conditions for both of these spaces to be closed under the Cauchy and the Dirichlet product. Further, a relationship between these products and the matrix transformations from these spaces to itself is established. These results extend of these of \textit{I. J. Maddox} and \textit{M. A. L. Willey} [Indian J. Math. 16, 23-31 (1974; Zbl 0352.46007)].