This is an announcement for the paper "Finding left inverses for classes of operators on l^p(Z^d) with some decay conditions" by Romain Tessera. Abstract: We study the left-invertibility of infinite matrices indexed by metric spaces with polynomial growth. In particular, we consider matrices with polynomial decay, indexed by discrete groups of polynomial growth. Under different conditions on the rows and the columns, we prove that being bounded-below in l^p for some p implies that there is a left-inverse which is bounded in l^q, for all q between 1 and infinity. Archive classification: math.FA Mathematics Subject Classification: 47B38, 47B37 Remarks: 33 pages The source file(s), thinop10.tex: 77101 bytes, is(are) stored in gzipped form as 0801.1532.gz with size 23kb. The corresponding postcript file has gzipped size 163kb. Submitted from: tessera@phare.normalesup.org The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.1532 or http://arXiv.org/abs/0801.1532 or by email in unzipped form by transmitting an empty message with subject line uget 0801.1532 or in gzipped form by using subject line get 0801.1532 to: math@arXiv.org.