Tensor splitting properties of n-inverse pairs of operators

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.