Abstract
We consider Banach algebras of infinite matrices defined in terms of a weight measuring the off-diagonal decay of the matrix entries. If a given matrix \(A\) is invertible as an operator on \(\ell ^2\) we analyze the decay of its inverse matrix entries in the case where the matrix algebra is not inverse closed in \({\mathcal B} (\ell ^2),\) the Banach algebra of bounded operators on \(\ell ^2.\) To this end we consider a condition on sequences of weights which extends the notion of GRS-condition. Finally we focus on the behavior of inverses of pseudodifferential operators whose Weyl symbols belong to weighted modulation spaces and the weights lack the GRS condition.
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