Tensor splitting properties of n-inverse pairs of operators

TL;DR

This paper investigates the structure of n-inverse pairs of operators on tensor product spaces, showing they originate from inverse pairs on individual spaces, and extends the results to n-symmetries using algebraic geometry techniques.

Contribution

It proves that n-inverse pairs on tensor products derive from inverse pairs on component spaces, confirming a conjecture and extending to n-symmetries with algebraic geometry methods.

Findings

n-inverse pairs on tensor products originate from pairs on individual spaces

The paper confirms a conjecture of the second author

Results extend to n-symmetries in tensor products

Abstract

In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l- and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and M\"uller, and proves a conjecture of the second named author. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor product as well.