Kruglov operator and operators defined by random permutations

TL;DR

This paper explores the relationship between the Kruglov operator and operators defined by random permutations, establishing their equivalence in boundedness within rearrangement-invariant spaces and showing the absence of a minimal space with the Kruglov property.

Contribution

It proves the equivalence of boundedness between the Kruglov operator and permutation-based operators in r.i. spaces and demonstrates that no minimal r.i. space possesses the Kruglov property.

Findings

Boundedness of the Kruglov operator is equivalent to uniform boundedness of permutation operators.

No minimal rearrangement-invariant space has the Kruglov property.

The results connect operator boundedness with geometric properties of function spaces.

Abstract

The Kruglov property and the Kruglov operator play an important role in the study of geometric properties of r.i. function spaces. We prove that the boundedness of the Kruglov operator in a r.i. space is equivalent to the uniform boundedness on this space of a sequence of operators defined by random permutations. It is shown also that there is no minimal r.i. space with the Kruglov property.