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We use this to show that there are pseudo-differential operators $\\operatorname{Op} (a)$ and $\\operatorname{Op} (b)$ which are inverses to each others, where $a\\in \\Gamma ^{(\\omega _0)}_*$ and $b\\in \\Gamma ^{(1/\\omega _0)}_*$.\n  We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. 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We use this to show that there are pseudo-differential operators $\\operatorname{Op} (a)$ and $\\operatorname{Op} (b)$ which are inverses to each others, where $a\\in \\Gamma ^{(\\omega _0)}_*$ and $b\\in \\Gamma ^{(1/\\omega _0)}_*$.\n  We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. 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