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DC Field | Value | Language |
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dc.contributor.author | Khokulan, M. | - |
dc.contributor.author | Ramakrishnan, R. | - |
dc.date.accessioned | 2023-11-08T04:39:20Z | - |
dc.date.available | 2023-11-08T04:39:20Z | - |
dc.date.issued | 2023 | - |
dc.identifier.citation | Khokulan M.; Ramakrishnan R. (2023), Controlled π² βframes 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. | en_US |
dc.identifier.uri | http://repository.kln.ac.lk/handle/123456789/26902 | - |
dc.description.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 (ππ»π ) and left quaternionic Hilbert space (ππ» πΏ). πΎ βframes are more general than ordinary frames in the sense that the lower frame bound only holds for the elements in the range of πΎ, where πΎis a bounded linear operator in ππ» πΏ. 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 πΎ βframe is introduced in left quaternionic Hilbert space along the lines of their real and complex counterparts and some of their properties were analysed. Let ππ» πΏ be a left quaternionic Hilbert space, πΎ β π΅(ππ» πΏ) and πΆ β πΊπΏ+(ππ» πΏ), where π΅(ππ» πΏ)is the set of all bounded linear operators and πΊπΏ+(ππ» πΏ) is the set of all positive bounded linear operators in ππ» πΏ with bounded inverse. A sequence of family π· = {ππ}πβπΌ in ππ» πΏ is called a πΆ β controlled πΎ β frame for ππ» πΏ if there exist constants π, π > 0 such that πβπΎβ πβ2 β€ Ξ£πβπΌ β¨ππβ©β¨πβ© β€ πβπβ2 , for all π β ππ» πΏ. First, we established a result that shows that any πΎ β frame is a controlled πΎ βframe under certain conditions. Let πΎ and πΆ be self -adjoint with πΆπΎ = πΎπΆ. If π· = {ππ}πβπΌ is a πΎ β frame for ππ» πΏ then π· = {ππ}πβπΌ is a πΆ β controlled πΎ β frame for ππ»πΏ. Then we derived a necessary and sufficient condition for a sequence to be a controlled πΎ β frame and we have shown that every πΆ β controlled πΎ β frame is a πΆβ1 β controlled πΎ β frame. Suppose that πΎ β π΅(ππ»πΏ). A sequence π· = {ππ}πβπΌ is a πΆ β controlledπΎ β frame for ππ»πΏ if and only if π (πΎ) β π (ππΆπ·), where π (πΎ) is the range of πΎ. Suppose thatπΆπΎ = πΎπΆ. If π· = {ππ}πβπΌ is a πΆ β controlled πΎ β frame for ππ»πΏ then π· = {ππ}πβπΌ is a πΆβ1 βcontrolled πΎ β frame for ππ»πΏ. Finally, we proved that the sum of two controlled πΎ β framesremains a controlled πΎ β frame under certain conditions in left quaternionic Hilbert space. LetπΆπΎ = πΎπΆ. Suppose that π· = {ππ}πβπΌ and πΉ = {ππ}πβπΌare πΆ β controlled πΎ β frames for ππ»πΏ with bounds π, π and πβ², πβ², respectively. If ππ·ππΉ β = πΆβ1πΎπΎβ , then {ππ + ππ}πβπΌ is also a πΆ β controlled πΎ β frame for ππ»πΏ. | en_US |
dc.publisher | Faculty of Science, University of Kelaniya Sri Lanka | en_US |
dc.subject | Frames, Quaternion, Quaternionic Hilbert space, πΎ βframes, Controlled frames | en_US |
dc.title | Controlled π² βframes in quaternionic setting | en_US |
Appears in Collections: | ICAPS 2023 |
Files in This Item:
File | Description | Size | Format | |
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ICAPS 2023 68.pdf | 213.81 kB | Adobe PDF | View/Open |
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