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Abstract. We introduce the notion of localizable operators with respect to frames and prove the boundedness of such operators on families of Banach spaces. This generalizes previous results for specific operators, such as pseudodifferential operators on modulation spaces. We also use this notion to provide sufficient conditions for the construction of frames which have the localization property.

The theorems of Balan, Casazza, Heil, and Landau concerning the removal of sets of positive density from frames with positive excess are extended using a more general, symmetric concept of localization of frames.

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We introduce a new definition of localization for frames which gets rid of the dependence on the indexing of the frames. Two main results of Gröchenig are extended to this definition, namely that the dual of a localized frame is also localized, and a frame localized with respect to another frame is a Banach frame for the associated family of Banach spaces. These results parallel the results of a more recent paper by Fornasier and Gröchenig.