Unbounded generalization of logarithmic representation of infinitesimal generators

TL;DR

This paper extends the logarithmic representation of infinitesimal generators to unbounded evolution operators, providing a rigorous framework crucial for nonlinear analysis and integrable systems.

Contribution

It introduces a generalized framework for representing unbounded infinitesimal generators, enabling better analysis of nonlinear evolution operators.

Findings

Generalized logarithmic representation applicable to unbounded operators

Framework for identifying infinitesimal generators with evolution operators

Facilitates rigorous treatment of nonlinear transforms in integrable systems

Abstract

The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms: e.g., transforms appearing in the theory of integrable systems.