Some results on g-frames in Hilbert spaces

Authors: ABDOLAZIZ ABDOLLAHI, ELHAM RAHIMI

Abstract: In this paper we show that every g-frame for a Hilbert space H can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. We also show that every g-frame can be written as a sum of two tight g-frames with g-frame bounds one or a sum of a g-orthonormal basis and a g-Riesz basis for H. We further give necessary and sufficient conditions on g-Bessel sequences {\Lambda_i \in L (H,H_i) : i \in J} and {\Gamma_i \in L(H,H_i): i \in J} and operators L_1, L_2 on H so that {\Lambda_iL_1+\Gamma_iL_2: i \in J} is a g-frame for H. We next show that a g-frame can be added to any of its canonical dual g-frame to yield a new g-frame.

Keywords: Frame, g-frame, g-orthonormal basis, tight g-frame, g-Bessel sequence

Full Text: PDF