2022 A Brief Exposition of the Space of Relatively Bounded Nonlinear Operators
P. Viswanathan
Author Affiliations +
Real Anal. Exchange 47(1): 121-134 (2022). DOI: 10.14321/realanalexch.47.1.1610537860


This note aims to present some rudimentary aspects of relatively bounded nonlinear (not necessarily linear) operators. To be precise, we prove that the set of all nonlinear operators between two normed linear spaces that are relatively bounded with respect to a fixed operator between these spaces itself is a normed linear space under a suitable norm. Furthermore, it is shown that if the codomain is a Banach space, then the space of all relatively bounded nonlinear operators is also complete. Some elementary results on the algebraic structure of the space of all nonlinear operators that are bounded with respect to a fixed operator are also recorded. The results reported in this note illustrate that with some predictable minor modifications, the fundamental classical results on the space of all bounded linear operators can be carried over to a more general setting. Here the word `general' is used in a double sense that the boundedness is generalized to relative boundedness and the restriction of linearity is dropped to include nonlinear maps.


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P. Viswanathan. "A Brief Exposition of the Space of Relatively Bounded Nonlinear Operators." Real Anal. Exchange 47 (1) 121 - 134, 2022. https://doi.org/10.14321/realanalexch.47.1.1610537860


Published: 2022
First available in Project Euclid: 13 June 2022

Digital Object Identifier: 10.14321/realanalexch.47.1.1610537860

Primary: 47L05 , 47L10
Secondary: 47L25

Keywords: Banach Algebra , Banach space , nonlinear operators , relatively bounded operators , space of operators

Rights: Copyright © 2022 Michigan State University Press


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Vol.47 • No. 1 • 2022
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