In the present paper, some sufficient and necessary conditions for two frames Φ = (ϕ n ) n and Ψ = (ψ n ) n under which they are approximately or generalized dual frames are determined depending on the properties of their analysis and synthesis operators. We also give a new characterization for approximately dual frames associated with a given frame and given operator by using of bounded operators. Among other things, we prove that if two frames Φ = (ϕ n ) n and Ψ = (ψ n ) n are close to each other, then we can find approximately dual frames Φ ad = (ϕ ad n ) n and Ψ ad = (ψ ad n ) n of them which are close to each other and T Φ U Φ ad = T Ψ U Ψ ad , where T Φ and T Ψ (resp. U Φ ad and U Ψ ad ) are the analysis operators (resp. synthesis operators) of the frames Φ and Ψ (resp. Φ ad and Ψ ad ), respectively. We then give some consequences on generalized dual frames. Finally, we apply these results to find some construction results for approximately dual frames for a given Gabor frame.Mathematics Subject Classification: Primary: 42C15; Secondary: 47A58.
In this paper we introduce the concept of von Neumann-Schatten Bessel multipliers in separable Banach spaces and obtain some of their properties. Finally, special attention is devoted to the study of invertible Hilbert-Schmidt frame multipliers. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for diagonalization of matrices in finite dimensional setting as well as for dual HS-frames.In particular, we show that a HS-frame is uniquely determined by the set of its dual HS-frames.2010 Mathematics Subject Classification. Primary 46C50, 65F15, 42C15; Secondary 41A58, 47A58.
Let A and B be Banach algebras and let T : B → A be a continuous homomorphism. Recently, we introduced a product M := A × T B, which is a strongly splitting Banach algebra extension of B by A. In the present paper, we characterize biprojectivity, approximate biprojectivity and biflatness of A × T B in terms of A and B. We also study some notions of amenability such as approximate amenability and pseudo-amenability of A × T B.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.