Isometries of generalized $log$-spaces

TL;DR

This paper characterizes isometries of generalized log-spaces of integrable functions, providing necessary and sufficient conditions based on Boolean algebra passports, and explores different types of log algebras.

Contribution

It introduces a comprehensive framework for understanding isometries in generalized log-spaces, including necessary and sufficient conditions using Boolean algebra passports.

Findings

Derived conditions for isometries in log-spaces with positive measures

Analyzed external, internal, and generalized log algebras separately

Established a complete characterization of isometries in these spaces

Abstract

In this paper studied isometries of $F$-spaces of integrable functions with logarithm. In particular, using passports of Boolean algebra, a necessary and sufficient condition of isometry $F$-spaces of integrable functions of logarithm with respect to strictly positive $\sigma$-finite measures is proved. In this work,external, internal and generalized $log$ algebras are considered separately.