Bicomplex Mergelyan's Approximation Theorem
In this paper we prove Area theorem, Biebarbach`s Theorem, Koebe Quarter Theorem and Mergelyan`s Approximation Theorem in the bicomplex framework.
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arXiv subjects
Romesh Kumar
Publications and source records attributed to Romesh Kumar.
Expansivity Property for Composition Operators on Orlicz -Lorentz Spaces
In this paper, we investige the concept of expansivity for composition operators on Orlicz-Lorentz spaces. We study necessary and sufficient conditions for expansivity, positive expansivity and uniformly expansivity for composition operators $C_{\tau}$ on $\mathbb{L}^{\varphi,h}(\mu)$. We extend the of results of [15] into Orlicz-Lorentz spaces
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Li-Yorke Chaos for Composition operators on Orlicz-Lorentz spaces
In this paper, we study the Li-Yorke chaotic composition operators on Orlicz-Lorentz space. In fact, necessary and sufficient conditions are given for Li-Yorke chaotic composition operator $C_{\tau}$ on $\mathbb{L}^{\varphi,h}(\mu)$. Further, we present the equivalent conditions for $C_{\tau}$ to be Li-Yorke chaotic. This paper's results are the generalization of results of [15] into Orlicz-Lorentz spaces.
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Li-Yorke and Expansive Composition Operators on Lorentz Spaces
In this paper, we investigate Li-Yorke composition operators and some of their variations on Lorentz spaces. Further, we also study expansive composition operators on these spaces. The work of the paper is essentially based on the work in [1], [9], [11], and [13].
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Hyperbolic valued random variables and conditional expectation
In this paper, we introduce the concept of hyperbolic valued random variables, their expectation and moments. We develop the hyperbolic analogue of Binomial and Poisson distributions. We study some of the properties of expectation on the basis of decomposition of a hyperbolic number into idempotent components. Finally we define conditional expectation of hyperbolic valued random variable and study some of its basic properties. Our random variable can take values which are zero divisors and this is the important part of this study.
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2-Normed D-Modules and Hahn-Banach Theorem for D-linear 2-functionals
In this paper we introduce the notion of D-valued 2-norm on hy- perbolic or D-valued modules. Further, we define D-linear 2-functional on these modules and consider some of their properties. We also establish the Hahn- Banach type extension theorem for D-linear 2-functionals.
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Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars
We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.
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Topological Bicomplex Modules
In this paper, we develop topological modules over the ring of bicomplex numbers. We discuss bicomplex convexivity, hyperbolic-valued seminorms and hyperbolic-valued Minkowski functionals in bicomplex modules. We also study the conditions under which topological bicomplex modules and locally bicomplex convex modules become hyperbolic normable and hyperbolic metrizable respectively.
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Maximal Ideals in a Bicomplex Algebra and Bicomplex Gelfand-Mazur Theorem
In this paper we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand-Mazur theorem for the bicomplex division Banach algebra.
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Bicomplex Weighted Hardy Spaces and Bicomplex C*-algebras
In this paper we study the bicomplex version of weighted Hardy spaces. Further, we describe reproducing kernels for the bicomplex weighted Hardy spaces. In particular, we generalize some results which holds for the classical weighted Hardy spaces. We also introduce the notion of bicomplex C*-algebra and discuss some of its properties including maximal ideals in BC.
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Bicomplex Linear Operators on Bicomplex Hilbert Spaces and Littlewood's Subordination Theorem
In this paper we study some basic properties of bicomplex linear operators on bicomplex Hilbert spaces. Further we discuss some applications of Hahn-Banach theorem on bicomplex Banach modules. We also introduce and discuss some bicomplex holomorphic function spaces and prove Littlewood's Subordination principle for bicomplex Hardy space.
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The Fundamental Theorems in the framework of Bicomplex Topological Modules
In this paper, we generalize the fundamental theorems of functional analysis to the framework of bicomplex topological modules.
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Boundedness, compactness and Schatten-class membership of weighted composition operators
The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces $\mathcal{A}^2 α(\mathbb{D})$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.
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Matrix multiplication operators on Banach function spaces
In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups for solving the abstract Cauchy problem.
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