Some results on G-frame operator in quaternionic setting

Abstract

G frames are a natural generalization of frames which cover many other extensions of frames. Quaternions are an extension of complex numbers from the twodimensional plane to four-dimensional space and form a non-commutative associative algebra. Due to the non-commutativity, there are two types of Hilbert spaces over quaternions, called right quaternionic Hilbert space and left quaternionic Hilbert space. In this research G-frame operator for G-frame in left quaternionic Hilbert space Vℍ L is introduced and some results of G-frame operator are presented and one can easily obtain these results on right quaternionic Hilbert space Vℍ R by the symmetry. Let Uℍ L and Vℍ L be two left quaternionic Hilbert spaces and {𝒱𝒱k: k ∈ 𝕀𝕀} ⊆ Vℍ L is a sequence of quaternionic Hilbert spaces. A family of sequence �Λk ∈ ℬ�Uℍ L ,𝒱𝒱k�: k ∈ 𝕀𝕀� is called generalized frame or simply G-frame for Uℍ L with respect to {𝒱𝒱k: k ∈ 𝕀𝕀} if there exist constants 0 < A ≤ B < ∞ such that A‖f‖2 ≤ ∑ ‖Λkf‖2 k∈𝕀𝕀 ≤ B‖f‖2, for all f ∈ Uℍ L , where A and B are G-frame bounds. We call �Λk ∈ ℬ�Uℍ L ,𝒱𝒱k�: k ∈ 𝕀𝕀� is a tight G-frame if A = B. If {Λk}k∈𝕀𝕀 is a G-frame in Uℍ L with G-frame operator SG if and only if AIop ≤ SG ≤ BIop and {Λk}k∈𝕀𝕀 is Gnormalized tight frame in Uℍ L if and only if SG = Iop, where Iop is an identity operator in Uℍ L . If SG is a G-frame operator for the G-frame {Λk}k∈𝕀𝕀 with frame bounds A and B in Uℍ L then B−1Iop ≤ SG−1 ≤ A−1Iop. Finally sequence of operator �Λ �k� k∈𝕀𝕀 (where Λ �k=ΛkSG−1) is G-frame for the quaternionic Hilbert space Uℍ L with frame bounds 1 B and 1 A. We have seen that a sequence of operators is a G-frame for the left quaternionic Hilbert space Vℍ L with frame bounds 1 B and 1 A.