Left inverses of matrices with polynomial decay |
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Authors: | Romain Tessera |
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Affiliation: | École Normale Supérieure de Lyon, CNRS UMR 5669, 46, allée d'Italie, Lyon, France |
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Abstract: | 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 ‖Afp‖?‖fp‖, 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. |
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Keywords: | Stability of Schur operators Left inverse for infinite matrices with off-diagonal decay |
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