Browsing by Author "Johnson, P.S."

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ALGEBRAIC PROOFS OF CHARACTERIZING REVERSE ORDER LAW FOR CLOSED RANGE OPERATORS IN HILBERT SPACES
(L.N. Gumilyov Eurasian National University, 2023) Athira, S.K.; Kamaraj, K.; Johnson, P.S.
We present more than 60 results, including some range inclusion results to characterize the reverse order law for the Moore-Penrose inverse of closed range Hilbert space operators. We use the basic properties of the Moore-Penrose inverse to prove the results. Some examples are also provided to illustrate failure cases of the reverse order law in an infinite-dimensional setting. © (2023). All Rights Reserved.
Anomaly Detection in Electric Powertrain System Software Behaviour
(Institute of Electrical and Electronics Engineers Inc., 2023) Vyas, A.; Ghorpade, V.; Kamble, S.; Johnson, P.S.; Kamath, A.; Rawat, K.
A software-in-loop (SIL) testing is a method of early testing of control software of a car in virtual environment. A system level testing is carried out on regular basis and it is important to see, if system is behaving as expected or unexpected. For unexpected behaviors, which test engineers not easily notice, modern techniques such as machine learning can give an advantage. This paper presents an application of machine learning algorithms that helps in identifying the abnormal patterns in time series data generated from electric powertrain system testing done in SIL environment for a Mercedes Benz Electric Car. Output of the SIL testing, results in time series data that is a collection of observations that are ordered chronologically and can be used to analyze trends, patterns, and changes over time. Anomaly detection in time series data is a process in machine learning that identifies data points, events, and observations that deviate from a dataset's normal behavior. By monitoring the expected and unexpected behavior of the electric powertrain system, anomaly detection can be a valuable tool for identifying potential issues. This study aims at coming up with an efficient process for anomaly detection in SIL. In order to get this process, various anomaly detection techniques are compared to detect a defined anomaly in time series data. Data pre-processing methods are also discussed before training the model. At the end, we conclude a best-fit method for identified anomaly. With finally identified method, a model was trained and used further in application. Â© 2023 IEEE.
APPROXIMATELY DUAL p-APPROXIMATE SCHAUDER FRAMES
(Austral Internet Publishing, 2023) Mahesh Krishna, K.M.; Johnson, P.S.
Approximately dual frame in Hilbert spaces was introduced by Christensen and Laugesen to overcome difficulties in constructing dual frames for a given Hilbert space frame. It becomes even more difficult in Banach spaces to construct duals. For this purpose, we introduce approximately dual frames for a class of approximate Schauder frames for Banach spaces and develop basic theory. Approximate dual for this subclass is completely characterized and its perturbation is also studied. © 2023 Austral Internet Publishing. All rights reserved
Class of bounded operators associated with an atomic system
(2015) Johnson, P.S.; Ramu, G.
K-frames, more general than the ordinary frames, have been introduced by Laura G?vru?a in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Using the frame operator, we find a class of bounded linear operators in which a given Bessel sequence is an atomic system for every member in the class.
Class of bounded operators associated with an atomic system
(Tamkang University editor@staff.tku.edu.tw, 2015) Johnson, P.S.; Ramu, G.
K-frames, more general than the ordinary frames, have been introduced by Laura G?vru?a in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Using the frame operator, we find a class of bounded linear operators in which a given Bessel sequence is an atomic system for every member in the class.
Closed EP and hypo-EP operators on Hilbert spaces
(Springer Science and Business Media B.V., 2022) Johnson, P.S.
A bounded linear operator A on a Hilbert space H is said to be an EP (hypo-EP) operator if ranges of A and A∗ are equal (range of A is contained in range of A∗) and A has a closed range. In this paper, we define EP and hypo-EP operators for densely defined closed linear operators on Hilbert spaces and extend results from bounded linear operator settings to (possibly unbounded) closed linear operator settings. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.
Commutators close to the identity in unital C*-algebras
(Springer, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
Let H be an infinite dimensional Hilbert space and B(H) be the C∗-algebra of all bounded linear operators on H, equipped with the operator-norm. By improving the Brown–Pearcy construction, Tao (J. Oper. Theory82(2) (2019) 369–382) extended the result of Popa (On commutators in properly infinite W∗-algebras, in: Invariant subspaces and other topics (1982) (Boston, Mass.: Birkhäuser, Basel)) which reads as: for each 0 < ε≤ 1 / 2 , there exist D, X∈ B(H) with ‖ [D, X] - 1 B(H)‖ ≤ ε such that ‖D‖‖X‖=O(log51ε), where [D, X] : = DX- XD. In this paper, we show that Tao’s result still holds for certain class of unital C*-algebras which include B(H) as well as the Cuntz algebra O2. © 2022, Indian Academy of Sciences.
Construction of Mercedes–Benz Frame in R n
(Springer, 2017) Parvathalu, B.; Johnson, P.S.
In this article, Mercedes–Benz (MB) frame having 3 vectors in R 2 is generalized to the space R n with n+ 1 vectors through a complete concrete method. A necessary and sufficient condition for a normed tight frame to be an MB frame is given and MB frame is explored with the help of diagram vectors. In a new approach, it has been proved that there is no MB frame in R n with more than n+ 1 vectors and there is always an equiangular tight frame for every n? 2 , using MB frame. © 2017, Springer India Pvt. Ltd.
Convergence of operators with closed range
(2019) Johnson, P.S.; Balaji, S.
Izumino has discussed a sequence of closed range operators (Tn) that converges to a closed range operator T on a Hilbert space to establish the convergence of Tn ? T for Moore-Penrose inverses. In general, if Tn ? T uniformly and each Tn has a closed range, then T need not have a closed range. Some sufficient conditions have been discussed on Tn and T such that T has a closed range whenever each Tn has a closed range. 2019 Khayyam Journal of Mathematics.
Convergence of operators with closed range
(Tusi Mathematical Research Group (TMRG) moslehian@memeber.ams.org, 2019) Johnson, P.S.; Balaji, S.
Izumino has discussed a sequence of closed range operators (Tn) that converges to a closed range operator T on a Hilbert space to establish the convergence of Tn† ? T† for Moore-Penrose inverses. In general, if Tn ? T uniformly and each Tn has a closed range, then T need not have a closed range. Some sufficient conditions have been discussed on Tn and T such that T has a closed range whenever each Tn has a closed range. © 2019 Khayyam Journal of Mathematics.
DILATION THEOREM FOR p-APPROXIMATE SCHAUDER FRAMES FOR SEPARABLE BANACH SPACES
(Palestine Polytechnic University, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space H is the image of a Riesz basis under an orthogonal projection from a separable Hilbert space H1 which contains H isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case. © Palestine Polytechnic University-PPU 2022.
DILATIONS OF LINEAR MAPS ON VECTOR SPACES
(Element D.O.O., 2022) Mahesh Krishna, K.M.; Johnson, P.S.
Dilation of linear maps on vector spaces has been recently introduced by Bhat, De, and Rakshit. This notion is a variant of vector space dilation introduced by Han, Larson, Liu, and Liu. We derive vector space versions of Wold decomposition, Halmos dilation, N-dilation, inter-twining lifting theorem and a variant of Ando dilation. It is noted further that unlike a kind of uniqueness of Halmos dilation of strict contractions on Hilbert spaces, vector space version of Halmos dilation cannot be characterized. © 2022, Element D.O.O.. All rights reserved.
DIRECT SUM OF LOWER SEMI-FRAMES IN HILBERT SPACES
(Canadian University of Dubai, 2025) Hemalatha, M.; Johnson, P.S.; Harikrishnan, P.K.
In this paper, structural properties of lower semi-frames in separable Hilbert spaces are explored with a focus on transformations under linear operators (may be unbounded). Also, the direct sum of lower semi-frames, providing necessary and sufficient conditions for the preservation of lower semiframe structure, is examined. © 2025, Canadian University of Dubai. All rights reserved.
Estimates of Norms on Krein Spaces
(Austral Internet Publishing, 2020) Athira, S.K.; Johnson, P.S.; Kamaraj, K.
Various norms can be defined on a Krein space by choosing different underlying fundamental decompositions. Some estimates of norms on Krein spaces are discussed and a few results in Bognar’s paper are generalized. © 2020. Austral Internet Publishing. All Rights Reserved.
Expansion of weak reconstruction sequences to approximate Schauder frames for Banach spaces
(World Scientific, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and weak reconstruction sequences which cannot be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand weak reconstruction sequences to approximate Schauder frames. © 2022 World Scientific Publishing Company.
Factorable weak operator-valued frames
(Birkhauser, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
The notion of operator-valued frames (OVFs) by Kaftal et al. (Trans Am Math Soc 361(12):6349–6385, 2009) and G-frames by Sun (J Math Anal Appl 322(1):437–452, 2006) do not include all generalizations of notions of frames for Hilbert spaces. For this purpose, we introduce the notion of weak operator-valued frames (weak OVFs) which covers all known generalizations of frames for Hilbert spaces. Theory of weak-OVFs is more demanding than the theory of OVFs due to the following: One, the weak frame operator may not factor and another, weak frame operator may not be positive. As a first step towards a reasonable theory, we impose factorability condition on weak frame operator. We then characterize and derive dilation results. Similarity and orthogonality notions are introduced and characterized. The notion is connected with groups as well as group-like unitary systems. We also derive stability results. © 2021, Tusi Mathematical Research Group (TMRG).
Frame operators of K-frames
(Springer Nature, 2016) Ramu, G.; Johnson, P.S.
A close relation between frames on a separable Hilbert space H and positive invertible bounded operators is known. In this paper, it is shown that for a bounded operator K on H, there is a relationship between K-frames and quotient operators. Results on K-frames have been proved through operator-theoretic results on quotient of bounded operators. © 2016, Sociedad Española de Matemática Aplicada.
Frames for Metric Spaces
(Birkhauser, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
We make a systematic study of frames for metric spaces. We prove that every separable metric space admits a metric Md-frame. Through Lipschitz-free Banach spaces we show that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. We derive some characterizations of metric frames. We also derive stability results for metric frames. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Frames for Operators in Banach Spaces
(2017) Geddavalasa, R.; Johnson, P.S.
A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces. 2017, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.
Frames for Operators in Banach Spaces
(Springer New York LLC barbara.b.bertram@gsk.com, 2017) Ramu, R.; Johnson, P.S.
A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces. © 2017, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.