In a previous note, the author proved that the algebra of Schur operators on
l^2 is not inverse-closed. When l^2=l^2(X) where X is a metric space, we 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 for a strictly polynomial weight is inverse-closed.
Here, we prove a result dealing with left-invertibility. Namely, if such an
operator is bounded below in l^p for some p, then it is bounded below for all
q, and it admits a left-inverse in the weighted Schur algebra.