Bicomplex Matrices and Operators: Jordan Forms, Invariant Subspace Lattice Diagrams, and Compact Operators

TL;DR

This paper generalizes linear algebra and operator theory concepts from complex to bicomplex spaces, introducing bicomplex vector spaces, Jordan forms, invariant subspace lattices, and compact operators, thereby expanding the mathematical framework.

Contribution

It defines bicomplex vector spaces and operators, including Jordan forms and invariant subspace lattices, for the first time, extending classical theory to bicomplex spaces.

Findings

Bicomplex vector spaces and their bases are defined via idempotent representation.

Bicomplex Jordan form and invariant subspace lattice are characterized.

Theory of compact operators is extended to bicomplex Banach and Hilbert spaces.

Abstract

This paper extends topics in linear algebra and operator theory for linear transformations on complex vector spaces to those on bicomplex Hilbert and Banach spaces. For example, Definition 3 for the first time defines a bicomplex vector space, its dimension, and its basis in terms of a corresponding vectorial idempotent representation, and the paper shows how an n by n bicomplex matrix's idempotent representation leads to its bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, the paper rigorously defines for the first time "bicomplex Banach and Hilbert" spaces, and then it expands the theory of compact operators on complex Banach and Hilbert spaces to those on bicomplex Banach and Hilbert spaces. In these ways, the paper shows that complex linear algebra and operator theory are not necessarily built upon the broadest and most natural set of scalars to study. Instead, when using the idempotent representation, such results surprisingly generalize in a straightforward way to the higher-dimensional case of bicomplex values.