International Journal For Multidisciplinary Research

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Recent Developments in Bicomplex Operator Theory

Author(s) Neetu Singh
Country India
Abstract Bicomplex operator theory has evolved from the idempotent reduction of bicomplex modules into paired complex spaces. However, recent research has focused on semigroups, compact operators, frames, reproducing kernels, tensor products, ergodic theory, and quantum-information structures. A critical mathematical review with theorem-level exposition is provided in this paper. Bicomplex-linear operators are decomposed as T=T₁e₁+T₂e₂, and their complex components determine their boundedness, compactness, adjoints, normality, self-adjointness, and unitarity. The reduced paired spectrum is distinguished from the full algebraic spectrum. The full spectrum is a union of cylinders and is typically unbounded, as λI−T is invertible only when both component resolvents exist. Consequently, the null cone alters the interpretation of spectral assertions. The following are derived: polynomial functional calculus, point spectra, componentwise resolvent formulas, and compact-normal spectral structure. The review subsequently examines the following developments from 2020–2025: bicomplex operator semigroups, frames, Bergman and Bloch spaces, mean ergodic theorems, matrix canonical forms, Wiener and Herglotz theories, neural operators, and a bicomplex Choi theorem. Although these advancements are substantial, a significant number of them are reductions to two complex theories that are coupled by bicomplex notation. The paper identifies the areas in which this reduction is complete and the areas in which genuinely bicomplex questions remain, particularly in the context of null-cone spectral geometry, C*-structures, unbounded operators, perturbation theory, and physically constrained quantum models.
bicomplex operator; idempotent decomposition; compact operator; adjoint; spectrum; resolvent; semigroup; bicomplex Hilbert space
Keywords bicomplex operator; idempotent decomposition; compact operator; adjoint; spectrum; resolvent; semigroup; bicomplex Hilbert space
Published In Volume 7, Issue 5, September-October 2025
Published On 2025-10-10
DOI https://doi.org/10.36948/ijfmr.2025.v07i05.85681
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CrossRef DOI prefix of IJFMR is 10.36948/ijfmr

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