Abstract:
Quaternions are an extension of complex numbers with one real and three imaginary parts. Quaternionic Hilbert space is a vector space under multiplication by quaternionic scalars, from the non-commutativity, the quaternionic Hilbert spaces are defined in two ways: left/right quaternionic Hilbert spaces. Frame is a spanning set of vectors, which are generally over complete (redundant) in a quaternionic Hilbert space. G- frames are natural generalization of frames and provide more choices on analyzing functions from frame expansion coefficients. In this research, the construction of G-frame is reported and relation between G-frame and canonical dual G-frame is established. Let 𝑈β
𝐿 and 𝑉β
𝐿 be left quaternionic Hilbert spaces and {𝒱𝑘βΆ𝑘β𝕀}β 𝑉β
𝐿 is a sequence of quaternionic Hilbert spaces. Let 𝔅(𝑈β
𝐿,𝒱𝑘) be the collection of all bounded linear operators from 𝑈β
𝐿 into𝒱𝑘. A family {Ξ𝑘β𝔅(𝑈β
𝐿,𝒱𝑘) βΆ𝑘β𝕀} is called a generalized frame or simply G-frame for 𝑈β
𝐿 with respect to {𝒱𝑘βΆ𝑘β𝕀} if there exist constants 0<𝐶β€𝐷<β such that 𝐶β𝜙β2β€Ξ£βΞ𝑘𝜙β2𝑘β𝕀β€𝐷β𝜙β2, for all 𝜙β𝑈β
𝐿, where 𝐶 and 𝐷 are G-frame bounds. G-frame operator 𝐹𝑔 can be defined as 𝐹𝑔𝜙=Ξ£Ξ𝑘β 𝑘β𝕀Ξ𝑘𝜙, for all 𝜙β𝑈β
𝐿, where Ξ𝑘β is the adjoint operator of Ξ𝑘. Frame operator 𝐹𝑔 is self adjoint, bounded and invertible. If {Ξ𝑘 βΆ𝑘β𝕀} be a G-frame for 𝑈β
𝐿 with respect to {𝒱𝑘βΆ𝑘β𝕀} and Ξ𝑘Μ=Ξ𝑘𝐹𝑔β1, then {Ξ𝑘Μ βΆ𝑘β𝕀} is a G-frame for 𝑈β
𝐿 with frame bounds 1𝐷 and 1𝐶. We call it the canonical dual G-frame of {Ξ𝑘 βΆ𝑘β𝕀}. Finally, we conclude that {Ξ𝑘 βΆ𝑘β𝕀} and {Ξ𝑘Μ βΆ𝑘β𝕀} are dual G-frames with respect to each other.