Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis

TL;DR

This paper develops a foundational functional analysis framework for bicomplex scalars, exploring bicomplex modules, norms, and matrices, and extends classical Schur analysis into the bicomplex setting.

Contribution

It introduces a rigorous theory of bicomplex functional analysis, including norms, modules, and a bicomplex Schur analysis, filling a gap in the mathematical understanding of bicomplex structures.

Findings

Hyperbolic-valued norms are more compatible with bicomplex modules.

A bicomplex version of classical Schur analysis is formulated.

Bicomplex matrices and linear functionals are systematically studied.

Abstract

With the goal of providing the foundations for a rigorous study of modules of bicomplex holomorphic functions, we develop a general theory of functional analysis with bicomplex scalars. Even though the basic properties of bicomplex number are well known and widely available, our analysis requires some more delicate discussion of the various structures which are hidden in the ring of bicomplex numbers. We study in particular matrices with bicomplex numbers, bicomplex modules, and inner products and norms in bicomplex modules. We consider two kinds of norms on bicomplex modules: a real-valued norm (as one would expect), and a hyperbolic-valued norm. Interestingly enough, while both norms can be used to build the theory of normed bicomplex modules, the hyperbolic-valued norm appears to be much better compatible with the structure of bicomplex modules. We also consider linear functionals on bicomplex numbers. To conclude we describe a bicomplex version of classical Schur analysis