[Paper Review] Orlicz Spaces with Bicomplex Scalars

Rajesh Kumar, Kanika Sharma+2 more|arXiv (Cornell University)|Jan 28, 2014

Holomorphic and Operator Theory3 citations

TL;DR

This paper introduces Orlicz spaces over bicomplex scalars using a hyperbolic-valued Luxemburg norm, generalizing classical Orlicz spaces to the bicomplex setting. It establishes foundational properties of these spaces, characterizes integral representations of $bb{D}$-valued convex functions, and proves that the spectrum of the unilateral shift operator on $l^{2}(\mathbb{BC})$ is the null cone $\mathcal{NC}$, extending functional analysis to non-associative hypercomplex algebras.

ABSTRACT

In this paper we define bicomplex Orlicz space with hyperbolic valued Luxemburg norm and discussed some of their properties. We have also partially characterize an integral representation of a $\mathbb{D}$-valued convex function. Further we have shown that the spectrum of the unilateral shift operator on $l^2(\mathbb{BC})$ is the null cone $\mathcal{NC}$.

Motivation & Objective

• To extend the theory of Orlicz spaces to bicomplex scalar-valued functions using hyperbolic-valued norms.
• To define and study $\mathbb{D}$-valued convex functions and their integral representations.
• To generalize Musielak-Orlicz function theory to the bicomplex setting with $\mathbb{D}$-valued modular functionals.
• To analyze spectral properties of linear operators, particularly the unilateral shift, in the context of $l^2(\mathbb{BC})$.
• To establish foundational results in bicomplex functional analysis by adapting complex and real Orlicz space theory.

Proposed method

• Defined bicomplex Orlicz spaces using a $\mathbb{D}$-valued Luxemburg norm derived from a $\mathbb{D}$-Musielak Orlicz function $\varphi_{\mathbb{D}}$.
• Introduced a $\mathbb{D}$-convex modular functional $I_{\varphi_{\mathbb{D}}}^{\mathbb{D}}(f) = \int_{\Omega_{\mathbb{D}}} \varphi_{\mathbb{D}}(t, |f(t)|_k) \, d\mu_{\mathbb{D}}$ for measurable bicomplex functions.
• Represented the modular functional as $I_{\varphi_{\mathbb{D}}}^{\mathbb{D}}(f) = I_{\varphi_{\mathbb{D}_1}}^{\mathbb{D}}(f_1)e_1 - I_{\varphi_{\mathbb{D}_2}}^{\mathbb{D}}(f_2)e_2$ using the hyperbolic basis $e_1, e_2$, enabling decomposition into complex components.
• Defined the $\mathbb{D}$-Luxemburg norm as $\|f\|_{\varphi_{\mathbb{D}}}^{\mathbb{D}} = \inf\{\lambda >^\prime 0 : I_{\varphi_{\mathbb{D}}}^{\mathbb{D}}(f/\alpha) \leq^\prime 1\}$, ensuring values in the complement of the null cone.
• Used the $\dagger$-conjugation to define invertibility and identify zero-divisors in $\mathbb{BC}$, crucial for spectral analysis.
• Analyzed the unilateral shift operator $S_{\mathbb{D}}^*$ on $l^2(\mathbb{BC})$, proving $\|S_{\mathbb{D}}^*\|_{\mathbb{D}} \leq^\prime 1$ and showing $0 \in \rho_{\mathbb{D}}(S_{\mathbb{D}}^*)$ via existence of the inverse right shift.

Experimental results

Research questions

• RQ1 How can Orlicz spaces be generalized to bicomplex scalars using a hyperbolic-valued Luxemburg norm?
• RQ2 What conditions ensure the modular functional $I_{\varphi_{\mathbb{D}}}^{\mathbb{D}}$ is well-defined and $\mathbb{D}$-convex for $\mathbb{D}$-Musielak Orlicz functions?
• RQ3 Can an integral representation be established for $\mathbb{D}$-valued convex functions in the bicomplex setting?
• RQ4 What is the spectrum of the unilateral shift operator on $l^2(\mathbb{BC})$ when equipped with a $\mathbb{D}$-valued norm?
• RQ5 How do the three conjugations and moduli on $\mathbb{BC}$ influence the structure of Orlicz spaces and operator theory?

Key findings

• The spectrum of the unilateral shift operator $S_{\mathbb{D}}^*$ on $l^2(\mathbb{BC})$ is the null cone $\mathcal{NC}$, as $0 \in \rho_{\mathbb{D}}(S_{\mathbb{D}}^*)$ and the inverse exists as the right shift.
• The $\mathbb{D}$-Luxemburg norm ensures $\|f\|_{\varphi_{\mathbb{D}}}^{\mathbb{D}}$ lies in the complement of the null cone $\mathcal{NC}$, preserving invertibility and norm structure.
• The modular functional decomposes as $I_{\varphi_{\mathbb{D}}}^{\mathbb{D}}(f) = I_{\varphi_{\mathbb{D}_1}}^{\mathbb{D}}(f_1)e_1 - I_{\varphi_{\mathbb{D}_2}}^{\mathbb{D}}(f_2)e_2$, enabling analysis via complex components.
• The $\mathbb{D}$-Musielak Orlicz function $\varphi_{\mathbb{D}}$ satisfies $\varphi_{\mathbb{D}}(t, |f(t)|_k) = \varphi_{\mathbb{D}_1}(t_1, |f_1(t_1)|)e_1 + \varphi_{\mathbb{D}_2}(t_2, |f_2(t_2)|)e_2$, ensuring compatibility with the hyperbolic decomposition.
• The operator $S_{\mathbb{D}}^*$ is bounded with $\|S_{\mathbb{D}}^*\|_{\mathbb{D}} \leq^\prime 1$, confirmed via norm estimation using the $k$-norm $|x_n|_k^2$.
• The inverse of $S_{\mathbb{D}}^*$ exists and is the right shift $S_{\mathbb{D}}^{*-1}(\xi_1, \xi_2, \xi_3) = (0, \xi_1, \xi_2, \xi_3)$, confirming $0$ is in the resolvent set.