International Journal For Multidisciplinary Research

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Volume 8 Issue 4
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Continuous Linear Operators on Bicomplex Multi-Normed Spaces

Author(s) Neetu Singh
Country India
Abstract In order to interact with a multi-norm, continuous bicomplex-linear operators must distinguish between ordinary boundedness and level-uniform multi-boundedness, despite the fact that they inherit two complex components. The corresponding operator theory is elaborated upon in this paper. Sequential continuity, ordinary boundedness, componentwise boundedness, continuity at zero, and continuity at every point are all demonstrated to be equivalent for bicomplex-linear maps between normed modules. The two component multi-bounds are the defining characteristic of multi-boundedness, which is demonstrated to imply ordinary boundedness. The converse is established for minimum multi-norms and other operator-stable structures. The hyperbolic operator multi-norm is defined by component infima, which prevents division by null-cone values. Uniform limits, closed graphs, finite-rank and compact operators, inverse operators, composition, and sums are addressed. The open mapping, bounded inverse, closed graph, and uniform boundedness theorems are derived from their complex components under bicomplex Banach hypotheses. Componentwise characterisation is employed to distinguish between strong and weak operator convergence. Four worked examples examine matrix operators, multiplication on continuous-function modules, shifts and compact diagonal operators on bicomplex sequence spaces, and rank-one Hilbert operators. The results demonstrate the precise classical equivalences that persist and the instances in which multi-level geometry imposes an additional condition. Compactness in nonminimum multi-norms, weak compactness, operator ideals, unbounded generators, and bicomplex multi-operator topologies are all open concerns.
Keywords continuous bicomplex operator; multi-boundedness; operator multi-norm; compact operator; closed graph; strong operator topology; weak operator topology
Published In Volume 8, Issue 4, July-August 2026
Published On 2026-08-25
DOI https://doi.org/10.36948/ijfmr.2026.v08i04.86616
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