Norm-Attainable Operators and Polynomials: Theory, Characterization, and Applications in Optimization and Functional Analysis
Mogoi N. Evans *
Department of Mathematics and Statistics, Kaimosi Friends University, P.O. BOX 385-50309, Kaimosi, Kenya.
Isaac O. Okwany
Department of Mathematics and Actuarial Science, Catholic University of Eastern Africa, P.O. BOX 62157-00200, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
Abstract
This research paper offers a comprehensive investigation into the concept of norm-attainability in Banach and Hilbert spaces. It establishes that norm-attainable operators exist if and only if the target space is a Banach space and that norm-attainable polynomials are inherently linear. In convex optimization scenarios, norm-attainable polynomials lead to unique global optima. The paper explores the norm of norm-attainable operators, revealing its connection to supremum norms. In Hilbert spaces, norm-attainable operators are self-adjoint. Additionally, it shows that in finite-dimensional spaces, all bounded linear operators are norm-attainable. The research also examines extremal polynomials and their relationship with derivative roots, characterizes optimal solutions in norm-attainable operator contexts, and explores equivalence between norm-attainable operators through invertible operators. In inner product spaces, norm-attainable polynomials are identified as constant. Lastly, it highlights the association between norm-attainable operators and convex optimization problems, where solutions lie on the unit ball's boundary. This paper offers a unified perspective with significant implications for functional analysis, operator theory, and optimization in various mathematical and scientific domains.
Keywords: Norm-attainable operators, optimization landscape, Banach and Hilbert spaces, extremal polynomials, convex optimization