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Bicomplex Hilbert Spaces and Their Applications

Author(s) Neetu Singh
Country India
Abstract Complex inner-product geometry is extended to modules over the bicomplex algebra by bicomplex Hilbert spaces. However, positivity is inherently hyperbolic, and zero divisors influence normalisation, independence, and spectral interpretation. This paper delineates a rigorous componentwise theory. In the hyperbolic cone, a bicomplex inner product is defined as a positive quadratic value that is obtained through idempotent conjugation. The componentwise square root is the norm that is associated. The bicomplex Cauchy–Schwarz and triangle inequalities, the projection theorem for closed submodules, and the Riesz representation theorem for continuous bicomplex-linear functionals are all established through detailed proofs. Under explicit component-dimension hypotheses, orthogonality, Pythagorean identities, orthogonal complements, Gram–Schmidt procedures, Bessel inequalities, Parseval identities, and basis expansions are derived. Division by a null-cone norm is invalid, and free orthonormal bases necessitate matched complex dimensions, as explained by the Gram–Schmidt analysis. Componentwise characterisations of self-adjoint, unitary, positive, compact, and normal operators are provided, and the normal spectral theorem is summarised in an idempotent form. Finite-dimensional modules, projections, L² spaces, and bicomplex reproducing-kernel Hilbert spaces are among the examples. The critical evaluation of applications to bicomplex quantum mechanics, Fourier representations, signal processing, frames, and analytic kernels is conducted. The theory provides precise mathematics for paired complex channels, whereas physical probabilities and genuinely coupled models necessitate additional axioms. Unbounded observables, C*-modules, frames with unequal component dimensions, tensor products, and spectral measures on bicomplex domains are all open problems.
Keywords bicomplex Hilbert space; hyperbolic norm; Cauchy–Schwarz inequality; projection theorem; Riesz representation; orthonormal basis; adjoint operator; quantum mechanics
Published In Volume 17, Issue 3, July-September 2026
Published On 2026-08-25
DOI https://doi.org/10.71097/IJSAT.v17.i3.11528
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