G- Frames in the quaternionic setting

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 𝑈ℍ 𝐿 and 𝑉ℍ 𝐿 be left quaternionic Hilbert spaces and {𝒱𝑘∶𝑘∈𝕀}⊆ 𝑉ℍ 𝐿 is a sequence of quaternionic Hilbert spaces. Let 𝔅(𝑈ℍ 𝐿,𝒱𝑘) be the collection of all bounded linear operators from 𝑈ℍ 𝐿 into𝒱𝑘. A family {Γ𝑘∈𝔅(𝑈ℍ 𝐿,𝒱𝑘) ∶𝑘∈𝕀} is called a generalized frame or simply G-frame for 𝑈ℍ 𝐿 with respect to {𝒱𝑘∶𝑘∈𝕀} if there exist constants 0<𝐶≤𝐷<∝ such that 𝐶‖𝜙‖2≤Σ‖Γ𝑘𝜙‖2𝑘∈𝕀≤𝐷‖𝜙‖2, for all 𝜙∈𝑈ℍ 𝐿, where 𝐶 and 𝐷 are G-frame bounds. G-frame operator 𝐹𝑔 can be defined as 𝐹𝑔𝜙=ΣΓ𝑘†𝑘∈𝕀Γ𝑘𝜙, for all 𝜙∈𝑈ℍ 𝐿, where Γ𝑘† is the adjoint operator of Γ𝑘. Frame operator 𝐹𝑔 is self adjoint, bounded and invertible. If {Γ𝑘 ∶𝑘∈𝕀} be a G-frame for 𝑈ℍ 𝐿 with respect to {𝒱𝑘∶𝑘∈𝕀} and Γ𝑘̃=Γ𝑘𝐹𝑔−1, then {Γ𝑘̃ ∶𝑘∈𝕀} is a G-frame for 𝑈ℍ 𝐿 with frame bounds 1𝐷 and 1𝐶. We call it the canonical dual G-frame of {Γ𝑘 ∶𝑘∈𝕀}. Finally, we conclude that {Γ𝑘 ∶𝑘∈𝕀} and {Γ𝑘̃ ∶𝑘∈𝕀} are dual G-frames with respect to each other.