Conditional expectations and interpolation of linear operators on ordered ideals between $L^1(0,1)$ and $L^\infty(0,1)$

TL;DR

This paper improves the Calderon-Ryff interpolation theorem and characterizes operators of conditional expectation that preserve order ideals in $L^1(0,1)$, providing insights into interpolation spaces between $L^1$ and $L^ f(0,1)$.

Contribution

It offers an improved version of the Calderon-Ryff theorem and describes conditions under which conditional expectation operators preserve order ideals.

Findings

Enhanced Calderon-Ryff interpolation theorem

Characterization of conditional expectation operators on order ideals

Identification of interpolation spaces between $L^1$ and $L^ f$

Abstract

The monograph contains the detailed exposition of the results obtained by the author during the last several years. In particular it contains an improvement of the well known Calderon - Ryff interpolation theorem and description of "verifying" operators of conditional mathematical expectations {if such an operator leaves an order ideal in $L^1% invariant, then the ideal is an interpolation space}.