Controlled ๐ฒ โframes in quaternionic setting
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Date
2023
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Faculty of Science, University of Kelaniya Sri Lanka
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 (๐๐ป๐
) and left quaternionic Hilbert space (๐๐ป ๐ฟ). ๐พ โframes are more general than ordinary frames in the sense that the lower frame bound only holds for the elements in the range of ๐พ, where ๐พis a bounded linear operator in ๐๐ป ๐ฟ. 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 ๐พ โframe is introduced in left quaternionic Hilbert space along the lines of their real and complex counterparts and some of their properties were analysed. Let ๐๐ป ๐ฟ be a left quaternionic Hilbert space, ๐พ โ ๐ต(๐๐ป ๐ฟ) and ๐ถ โ ๐บ๐ฟ+(๐๐ป ๐ฟ), where ๐ต(๐๐ป ๐ฟ)is the set of all bounded linear operators and ๐บ๐ฟ+(๐๐ป ๐ฟ) is the set of all positive bounded linear operators in ๐๐ป ๐ฟ with bounded inverse. A sequence of family ๐ท = {๐๐}๐โ๐ผ in ๐๐ป ๐ฟ is called a ๐ถ โ controlled ๐พ โ frame for ๐๐ป ๐ฟ if there exist constants ๐, ๐ > 0 such that ๐โ๐พโ ๐โ2 โค ฮฃ๐โ๐ผ โจ๐๐โฉโจ๐โฉ โค ๐โ๐โ2 , for all ๐ โ ๐๐ป ๐ฟ. First, we established a result that shows that any ๐พ โ frame is a controlled ๐พ โframe under certain conditions. Let ๐พ and ๐ถ be self -adjoint with ๐ถ๐พ = ๐พ๐ถ. If ๐ท = {๐๐}๐โ๐ผ is a ๐พ โ frame for ๐๐ป ๐ฟ then ๐ท = {๐๐}๐โ๐ผ is a ๐ถ โ controlled ๐พ โ frame for ๐๐ป๐ฟ. Then we derived a necessary and sufficient condition for a sequence to be a controlled ๐พ โ frame and we have shown that every ๐ถ โ controlled ๐พ โ frame is a ๐ถโ1 โ controlled ๐พ โ frame. Suppose that ๐พ โ ๐ต(๐๐ป๐ฟ). A sequence ๐ท = {๐๐}๐โ๐ผ is a ๐ถ โ controlled๐พ โ frame for ๐๐ป๐ฟ if and only if ๐
(๐พ) โ ๐
(๐๐ถ๐ท), where ๐
(๐พ) is the range of ๐พ. Suppose that๐ถ๐พ = ๐พ๐ถ. If ๐ท = {๐๐}๐โ๐ผ is a ๐ถ โ controlled ๐พ โ frame for ๐๐ป๐ฟ then ๐ท = {๐๐}๐โ๐ผ is a ๐ถโ1 โcontrolled ๐พ โ frame for ๐๐ป๐ฟ. Finally, we proved that the sum of two controlled ๐พ โ framesremains a controlled ๐พ โ frame under certain conditions in left quaternionic Hilbert space. Let๐ถ๐พ = ๐พ๐ถ. Suppose that ๐ท = {๐๐}๐โ๐ผ and ๐น = {๐๐}๐โ๐ผare ๐ถ โ controlled ๐พ โ frames for ๐๐ป๐ฟ with bounds ๐, ๐ and ๐โฒ, ๐โฒ, respectively. If ๐๐ท๐๐น โ = ๐ถโ1๐พ๐พโ , then {๐๐ + ๐๐}๐โ๐ผ is also a ๐ถ โ controlled ๐พ โ frame for ๐๐ป๐ฟ.
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Keywords
Frames, Quaternion, Quaternionic Hilbert space, ๐พ โframes, Controlled frames
Citation
Khokulan M.; Ramakrishnan R. (2023), Controlled ๐ฒ โframes in quaternionic setting, Proceedings of the International Conference on Applied and Pure Sciences (ICAPS 2023-Kelaniya) Volume 3, Faculty of Science, University of Kelaniya Sri Lanka. Page 68.