Multipliers of H 1 and Hankel Matrices
James Hedlund · Proceedings of the American Mathematical Society · 1969
A sequenceX = {X(n) } of complex numbers is said to be a multiplier of H' into the sequence space 11 if Xf= {X(n)f(n) } Ell' for every f(z) E= f(n)znCHl. The space of all such multipliers is denoted (HI', 11). The only important known result about (H', 11) is the inequality of Hardy [5, p. 236]: {1/(n+1)} E(H1, 11). Other similar multiplier spaces have been completely characterized: an elementary sufficient condition for (H', 12), proved by Hardy and Littlewood [4], is also necessary, and the spaces (H', l) for 2 ? q ? ?O can be described similarly. Also (HP, 11) has been determined recently for O<p<1 by Duren and Shields [3 ], while (H2, 11) is trivial. (H', 11) is a more interesting space, due to its equivalent formulations; perhaps in consequence it seems more difficult to determine. In this paper we give an alternate description of (H', 11) in terms of Hankel matrices and use this matrix description to derive somne necessary and some sufficient conditions.