# Decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces

## TL;DR

This paper introduces decomposable operators between different $L^p$-direct integrals of Banach spaces, generalizing composition operators, and establishes their boundedness criteria.

## Contribution

It defines a new class of decomposable operators acting between distinct $L^p$-direct integrals and provides necessary and sufficient conditions for their boundedness.

## Key findings

- Decomposable operators generalize composition operators.
- Boundedness conditions are characterized precisely.
- The framework extends operator theory in Banach space integrals.

## Abstract

The notion of decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces is introduced. We show that these operators generalize the composition operator, in sense that a mapping is replaced by a binary relation. The necessary and sufficient conditions for the boundedness of those operators are the main results of the paper.