Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on 2 (namely operators bounded on both 1 and ∞) is not inverse-closed. When 2 = 2(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A ∈ Aω(X) satisfies Af p f p, for some 1 p ∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in 2. The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X = Zd , under additional conditions on the decay. © 2010 Elsevier Inc. All rights reserved.