Dilation of a family of g-frames

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, Islamic Republic of Iran


In this paper, we first discuss about canonical dual of g-frame ΛP = {ΛiP B(H, Hi) : i I}, where Λ = {Λi B(H, Hi) : i I} is a g-frame for a Hilbert space H and P is the orthogonal projection from H onto a closed subspace M. Next, we prove that, if Λ = {Λi B(H, Hi) : i I} and Θ = {Θi B(K, Hi) : i I} be respective g-frames for non zero Hilbert spaces H and K, and Λ and Θ are unitarily equivalent (similar), then Λ and Θ can not be weakly disjoint. On the other hand, we study dilation property for g-frames and we show that two g-frames for a Hilbert space have dilation property, if they are disjoint, or they are similar, or one of them is similar to a dual g-frame of another one. We also prove that a family of g-frames for a Hilbert space has dilation property, if all the members in that family have the same deficiency.


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