Browsing by Author "Mahesh Krishna, K.M."
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Item 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 reservedItem 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.Item 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.Item 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.Item 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.Item 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).Item 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.Item MULTIPLIERS FOR OPERATOR-VALUED BESSEL SEQUENCES AND GENERALIZED HILBERT-SCHMIDT CLASSES(Korean Society for Computational and Applied Mathematics, 2022) Mahesh Krishna, K.M.; Johnson, P.S.; Mohapatra, R.N.In 1960, Schatten studied operators of the form Σ∞ n=1 λn(xn⊗yn), where {xn}n and {yn}n are orthonormal sequences in a Hilbert space, and {λn}n ∈ ℓ∞(ℕ). Balazs generalized some of the results of Schatten in 2007. In this paper, we further generalize results of Balazs by studying the operators of the form Σ∞ n=1 λn(A∗ nxn ⊗ B∗ n yn), where {An}n and {Bn}n are operator-valued Bessel sequences, {xn}n and {yn}n are sequences in the Hilbert space such that {∥xn∥∥yn∥}n ∈ ℓ∞(ℕ). We also generalize the class of Hilbert-Schmidt operators studied by Balazs. © 2022 KSCAM.Item Operator-valued p-approximate schauder frames(Ramanujan Mathematical Society, 2023) Mahesh Krishna, K.M.; Johnson, P.S.We give an operator-algebraic treatment of the theory of p-approximate Schuader frames which includes the theory of operator-valued frames studied by Kaftal, Larson, and Zhang [Trans. AMS., 2009], G-frames studied by Sun [J. Math. Anal. Appl., 2006], factorable weak operator-valued frames studied by Krishna and Johnson [Ann. Funct. Anal., 2022] and p-approximate Schauder frames studied by Krishna and Johnson [J. Pseudo-Differ. Oper. Appl., 2021] as particular cases. We show that a sufficiently rich theory can be developed even for Banach spaces. We achieve this by defining various concepts and characterizations in Banach spaces. These include duality, approximate duality, equivalence, orthogonality and stability. © 2023 Ramanujan Mathematical Society. All rights reserved.Item PERTURBATION OF P-APPROXIMATE SCHAUDER FRAMES FOR SEPARABLE BANACH SPACES(Poincare Publishers, 2021) Mahesh Krishna, K.M.; Johnson, P.S.Paley-Wiener theorems for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach spaces. We show that our result gives Paley-Wiener theorem for frames for Hilbert spaces. © Poincare Publishers.Item THE NONCOMMUTATIVE l1-l2 INEQUALITY FOR HILBERT C*-MODULES AND THE EXACT CONSTANT(Kyungnam University Press, 2022) Mahesh Krishna, K.M.; Johnson, P.S.Let A be a unital C*-algebra. Then it follows that (Formula Presented) By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain n-tuple x = (Formula Presented), we give a method to compute a positive element cx in the C*-algebra A such that the equality (Formula Presented) holds. We give an application for the integral of Kasparov. We also derive a formula for the exact constant for the continuous l1 - l2 inequality. © 2022. Kyungnam University PressItem Towards characterizations of approximate Schauder frame and its duals for Banach spaces(Birkhauser, 2021) Mahesh Krishna, K.M.; Johnson, P.S.Characterizations for a frame and its duals are known for separable Hilbert spaces. In this paper, we characterize a class of approximate Schauder frame and its duals for separable Banach spaces. We also give an operator-theoretic characterization for similarity of ASFs. Our results encode the results of Holub, Li, Balan, Han, and Larson. We also address orthogonality of ASFs. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.