Basics Of Functional Analysis With Bicomplex Sc... Jun 2026

Functional analysis with bicomplex scalars is not a mere generalization for its own sake. By leveraging the idempotent decomposition, it transforms into two independent complex theories linked by a common real norm. This duality reveals new spectral geometries and provides natural algebraic structures for problems involving two complex parameters. As research accelerates, bicomplex methods are finding their place in signal processing, relativity, and quantum theory. The basics are now established—the next decade will build the cathedral.

The spectrum of a bicomplex linear operator is not a subset of (\mathbbBC) in a simple way. Because of zero divisors, the resolvent set must avoid the non-invertible elements. The decomposes as: [ \sigma_\mathbbBC(T) = \sigma(T_1) \mathbfe_1 + \sigma(T_2) \mathbfe_2 ] where (\sigma(T_k)) are classical complex spectra. This "bicomplex spectrum" is a set of hyperbolic numbers — lines in (\mathbbBC) — leading to new spectral phenomena like "spectral zones" rather than discrete points. Basics of Functional Analysis with Bicomplex Sc...

A on a module $X$ is a function $| \cdot | Functional analysis with bicomplex scalars is not a