Controlled �� −frames in quaternionic setting

Abstract

Quaternion is an extension of complex numbers from the two-dimensional plane to fourdimensional space and forms non-commutative division algebra. A feature of quaternion is that the multiplication of two quaternions is non-commutative, from the non-commutativity the quaternionic Hilbert spaces are defined in two ways such as right quaternionic Hilbert space (𝑉𝐻𝑅) and left quaternionic Hilbert space (𝑉𝐻 𝐿). 𝐾 −frames are more general than ordinary frames in the sense that the lower frame bound only holds for the elements in the range of 𝐾, where 𝐾is a bounded linear operator in 𝑉𝐻 𝐿. Controlled frame is one of the newest generalizations of the frame which has been introduced to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this research, the notion of a controlled 𝐾 −frame is introduced in left quaternionic Hilbert space along the lines of their real and complex counterparts and some of their properties were analysed. Let 𝑉𝐻 𝐿 be a left quaternionic Hilbert space, 𝐾 ∈ 𝐵(𝑉𝐻 𝐿) and 𝐶 ∈ 𝐺𝐿+(𝑉𝐻 𝐿), where 𝐵(𝑉𝐻 𝐿)is the set of all bounded linear operators and 𝐺𝐿+(𝑉𝐻 𝐿) is the set of all positive bounded linear operators in 𝑉𝐻 𝐿 with bounded inverse. A sequence of family 𝛷 = {𝜑𝑘}𝑘∈𝐼 in 𝑉𝐻 𝐿 is called a 𝐶 − controlled 𝐾 − frame for 𝑉𝐻 𝐿 if there exist constants 𝑚, 𝑀 > 0 such that 𝑚‖𝐾†𝜑‖2 ≤ Σ𝑘∈𝐼 ⟨𝜑𝑘⟩⟨𝜑⟩ ≤ 𝑀‖𝜑‖2 , for all 𝜑 ∈ 𝑉𝐻 𝐿. First, we established a result that shows that any 𝐾 − frame is a controlled 𝐾 −frame under certain conditions. Let 𝐾 and 𝐶 be self -adjoint with 𝐶𝐾 = 𝐾𝐶. If 𝛷 = {𝜑𝑘}𝑘∈𝐼 is a 𝐾 − frame for 𝑉𝐻 𝐿 then 𝛷 = {𝜑𝑘}𝑘∈𝐼 is a 𝐶 − controlled 𝐾 − frame for 𝑉𝐻𝐿. Then we derived a necessary and sufficient condition for a sequence to be a controlled 𝐾 − frame and we have shown that every 𝐶 − controlled 𝐾 − frame is a 𝐶−1 − controlled 𝐾 − frame. Suppose that 𝐾 ∈ 𝐵(𝑉𝐻𝐿). A sequence 𝛷 = {𝜑𝑘}𝑘∈𝐼 is a 𝐶 − controlled𝐾 − frame for 𝑉𝐻𝐿 if and only if 𝑅(𝐾) ⊆ 𝑅(𝑇𝐶𝛷), where 𝑅(𝐾) is the range of 𝐾. Suppose that𝐶𝐾 = 𝐾𝐶. If 𝛷 = {𝜑𝑘}𝑘∈𝐼 is a 𝐶 − controlled 𝐾 − frame for 𝑉𝐻𝐿 then 𝛷 = {𝜑𝑘}𝑘∈𝐼 is a 𝐶−1 −controlled 𝐾 − frame for 𝑉𝐻𝐿. Finally, we proved that the sum of two controlled 𝐾 − framesremains a controlled 𝐾 − frame under certain conditions in left quaternionic Hilbert space. Let𝐶𝐾 = 𝐾𝐶. Suppose that 𝛷 = {𝜑𝑘}𝑘∈𝐼 and 𝛹 = {𝜓𝑘}𝑘∈𝐼are 𝐶 − controlled 𝐾 − frames for 𝑉𝐻𝐿 with bounds 𝑚, 𝑀 and 𝑚′, 𝑀′, respectively. If 𝑇𝛷𝑇𝛹 † = 𝐶−1𝐾𝐾†, then {𝜑𝑘 + 𝜓𝑘}𝑘∈𝐼 is also a 𝐶 − controlled 𝐾 − frame for 𝑉𝐻𝐿.