Matrix representations for linear transformations on series analytic in the unit disc

Abstract

Let S be the space of all complex sequences A such that if £ is a complex number and | z | < 1 then ^ Anzn converges. We present three characterizations of the linear transformations from S to S which have matrix representations. We also characterize the linear transformations from S to the bounded sequences (or to the convergent sequences) which have matrix representations. The characterizations are in terms of natural topologies for the spaces. These results are a blend of Kothe and Toeplitz' much quoted study [3] of complex sequence spaces, Haplanov's beautiful characterization [1] of those matrices which transform S to S, and some rather natural norms for S which have been used by V. Ganapathy Iyer [2] in his study of entire functions. Kothe and Toeplitz study complex sequence spaces, matrices, and linear transformations having a kind of continuity which is independent of norms. A space is said to be normal provided that if x is in the space and | yn | ^ | xn |, n = 0,1, , then y is also in the space. Kothe and Toeplitz show that a continuous linear transformation from a normal space to a normal space has a matrix representation, and conversely, provided that each space contains all the finite sequences. Our space S is normal and the space of bounded sequences is also normal. The continuity criteria used in our theorems (statement (2) in each) are special cases of the continuity condition of Kothe and Toeplitz. It follows from their work that the existence of a matrix for a linear transformation L is necessary and sufficient for L to have analytic continuity (see definition below). Given a matrix transformation from S to S and a norm Nr (0 < r < 1: if A is in S, Nr(A) = Σ*U \AP τp) for S, Haplanov's theorem provides another such norm NB such that the transformation is continuous from the normed linear space {S, Nr} to {S, NR}. Finally, to complete Theorem 1, each linear transformation which is continuous relative to some such pair of norms is represented by a matrix, even though S is complete with respect to neither of the norms. Our second theorem is like the first: the transformations are from S to the bounded sequences (or convergent sequences).