Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ℓ² (namely operators bounded on both ℓ1 and ℓ∞) is not inverse-closed. When ℓ²=ℓ²(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 A_ω(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 AεA_ω(X) satisfies ∥Af∥_p{succeeds or equal to}∥f∥_p, for some 1≤p≤∞, then it admits a left-inverse in A_ω(X). The main difficulty here is to obtain the above inequality in ℓ². 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=Z^d, under additional conditions on the decay.