In this paper we prove the Runges approximation theorem for bicomplex numbers.

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.

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_τ on 𝕃^φ,h(μ). Further, we present the equivalent conditions for C_τ to be Li-Yorke chaotic. This paper's results are the generalization of results of [15] into Orlicz-Lorentz spaces.

In this paper we prove Area theorem, Biebarbach`s Theorem, Koebe Quarter Theorem and Mergelyan`s Approximation Theorem in the bicomplex framework.

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].

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

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.

In this paper we have defined some product type hyperbolic convex function and product type hyperbolic Young function and prove some results based on this. We also characterize an integral representation of the product type hyperbolic convex functions in this paper.

In this paper, we introduce the concept of hyperbolic valued measure and discuss some of its basic properties. We develop the hyperbolic analogue of some fundamental theorems of measure theory. We also define distribution function of a D-random variable and studied conditional expectation when conditioning is based on random variable.Finally we develop the hyperbolic analogue of Fundamental Law of Probability which plays a basic role in the field of probability theory. AMS subject classification:

In this paper, we characterize the essentially algebraic composition operators on Lorentz sequence spaces with a weight. The techniques used in the proofs of the results of this paper are essentially the same as in the work of B\(\ddot{o}\)ttcher and Heidler (Integr Eqn Oper Theory 15:389–411, 1992 [2], St. Petersburg Math J 5:1099–1119, 1994 [3]).

In this paper, we develop topological modules over the ring of bicomplex numbers. We discuss bicomplex convexity, 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.

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.

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.

In this paper we study some basic properties of bicomplex linear operators on bicomplex Hilbert spaces. Further, we introduce the notion of bicomplex quotient modules and annihilators of submodules of a bicomplex module. We also introduce and discuss some bicomplex holomorphic function spaces and prove Littlewood's Subordination Principle for bicomplex Hardy space.

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.

In this paper we characterize multiplication operators induced by operator valued maps on Banach function spaces. We also study multiplication semigroups and stability of these operators.

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.

In this paper, we generalize the fundamental theorems of functional analysis to the framework of bicomplex topological modules.

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 $${\mathcal{A}^2_\alpha(\mathbb{D})}$$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.