Bounded Linear Operators on Hyperbolic-Valued Multi-Normed Spaces

Authors: Neetu Singh

DOI: https://doi.org/10.37082/IJIRMPS.v12.i6.233192

Short DOI: https://doi.org/

Country: India

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Abstract: This paper introduces a componentwise operator theory for hyperbolic-valued multi-normed spaces. A quotient formula for an operator norm is generally hazardous when the denominator is in the null cone, as hyperbolic scalars contain zero divisors. The multi-bound is defined as a pair of real infimal constants, and the analysis decomposes each hyperbolic module and operator through the orthogonal idempotents. The multi-boundedness of the two component operators is demonstrated to be equivalent to multi-boundedness, which implies ordinary boundedness and continuity. Norm properties, composition estimates, kernels, ranges, inverses, and quotient-induced operators are derived. Hyperbolic statements are generated by the recombination of component versions of the open mapping, bounded inverse, closed graph, and uniform boundedness principles under Banach hypotheses. The resolvent is defined as the Cartesian product of the component resolvents. As a result, the full hyperbolic spectrum is a union of spectral cylinders rather than a coupled set of component spectral values. To differentiate between these concepts, a reduced idempotent spectrum is introduced. Diagonal, projection, and shift operators are among the examples that have been implemented. The results provide a precise indication of which classical arguments are able to withstand idempotent reduction and which necessitate additional multi-norm compatibility. Compact multi-operators, Fredholm theory, spectral radii, weak operator topologies, and hyperbolic operator inequalities are among the remaining issues.

Keywords: hyperbolic-linear operator; multi-bounded operator; operator multi-norm; closed graph theorem; uniform boundedness; hyperbolic spectrum; zero divisor

Paper Id: 233192

Published On: 2024-12-05

Published In: Volume 12, Issue 6, November-December 2024

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