Infinite Dimensional Bicomplex Spectral Decomposition Theorem

TL;DR

This paper extends the Spectral Decomposition Theorem to infinite dimensional bicomplex Hilbert spaces, introducing key operator concepts within this framework and relating them to classical Hilbert space theory.

Contribution

It develops a bicomplex version of the spectral decomposition theorem and introduces operator concepts specific to bicomplex Hilbert spaces.

Findings

Established a bicomplex spectral decomposition theorem.

Defined bounded linear, orthogonal complement, and compact operators in bicomplex Hilbert spaces.

Connected bicomplex operator theory with classical Hilbert space concepts.

Abstract

This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space $M'$ imbedded in any bicomplex Hilbert space $M$.