Controlled ๐‘ฒ โˆ’frames in quaternionic setting

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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 (๐‘‰๐ป๐‘…) and left quaternionic Hilbert space (๐‘‰๐ป ๐ฟ). ๐พ โˆ’frames are more general than ordinary frames in the sense that the lower frame bound only holds for the elements in the range of ๐พ, where ๐พis a bounded linear operator in ๐‘‰๐ป ๐ฟ. 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 ๐พ โˆ’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 ๐‘‰๐ป ๐ฟ be a left quaternionic Hilbert space, ๐พ โˆˆ ๐ต(๐‘‰๐ป ๐ฟ) and ๐ถ โˆˆ ๐บ๐ฟ+(๐‘‰๐ป ๐ฟ), where ๐ต(๐‘‰๐ป ๐ฟ)is the set of all bounded linear operators and ๐บ๐ฟ+(๐‘‰๐ป ๐ฟ) is the set of all positive bounded linear operators in ๐‘‰๐ป ๐ฟ with bounded inverse. A sequence of family ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ in ๐‘‰๐ป ๐ฟ is called a ๐ถ โˆ’ controlled ๐พ โˆ’ frame for ๐‘‰๐ป ๐ฟ if there exist constants ๐‘š, ๐‘€ > 0 such that ๐‘šโ€–๐พโ€ ๐œ‘โ€–2 โ‰ค ฮฃ๐‘˜โˆˆ๐ผ โŸจ๐œ‘๐‘˜โŸฉโŸจ๐œ‘โŸฉ โ‰ค ๐‘€โ€–๐œ‘โ€–2 , for all ๐œ‘ โˆˆ ๐‘‰๐ป ๐ฟ. First, we established a result that shows that any ๐พ โˆ’ frame is a controlled ๐พ โˆ’frame under certain conditions. Let ๐พ and ๐ถ be self -adjoint with ๐ถ๐พ = ๐พ๐ถ. If ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ is a ๐พ โˆ’ frame for ๐‘‰๐ป ๐ฟ then ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ is a ๐ถ โˆ’ controlled ๐พ โˆ’ frame for ๐‘‰๐ป๐ฟ. Then we derived a necessary and sufficient condition for a sequence to be a controlled ๐พ โˆ’ frame and we have shown that every ๐ถ โˆ’ controlled ๐พ โˆ’ frame is a ๐ถโˆ’1 โˆ’ controlled ๐พ โˆ’ frame. Suppose that ๐พ โˆˆ ๐ต(๐‘‰๐ป๐ฟ). A sequence ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ is a ๐ถ โˆ’ controlled๐พ โˆ’ frame for ๐‘‰๐ป๐ฟ if and only if ๐‘…(๐พ) โŠ† ๐‘…(๐‘‡๐ถ๐›ท), where ๐‘…(๐พ) is the range of ๐พ. Suppose that๐ถ๐พ = ๐พ๐ถ. If ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ is a ๐ถ โˆ’ controlled ๐พ โˆ’ frame for ๐‘‰๐ป๐ฟ then ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ is a ๐ถโˆ’1 โˆ’controlled ๐พ โˆ’ frame for ๐‘‰๐ป๐ฟ. Finally, we proved that the sum of two controlled ๐พ โˆ’ framesremains a controlled ๐พ โˆ’ frame under certain conditions in left quaternionic Hilbert space. Let๐ถ๐พ = ๐พ๐ถ. Suppose that ๐›ท = {๐œ‘๐‘˜}๐‘˜โˆˆ๐ผ and ๐›น = {๐œ“๐‘˜}๐‘˜โˆˆ๐ผare ๐ถ โˆ’ controlled ๐พ โˆ’ frames for ๐‘‰๐ป๐ฟ with bounds ๐‘š, ๐‘€ and ๐‘šโ€ฒ, ๐‘€โ€ฒ, respectively. If ๐‘‡๐›ท๐‘‡๐›น โ€  = ๐ถโˆ’1๐พ๐พโ€ , then {๐œ‘๐‘˜ + ๐œ“๐‘˜}๐‘˜โˆˆ๐ผ is also a ๐ถ โˆ’ controlled ๐พ โˆ’ frame for ๐‘‰๐ป๐ฟ.

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Keywords

Frames, Quaternion, Quaternionic Hilbert space, ๐พ โˆ’frames, Controlled frames

Citation

Khokulan M.; Ramakrishnan R. (2023), Controlled ๐‘ฒ โˆ’frames 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.

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