Matrices which commute with Menon operators

If A is a nonnegative square matrix and X is a vector, then the Menon operator associated with A, denoted by TA, is defined by (TAX)i = (n/sigma over j=1 (A) ji (n/sigma over k=1 (A) jk (X)k)-1)-1. A close relation 18 known to exist between doubly stochastic matrices and Menon operators. The following problem is investigated: If each of E and F is a matrix, when is ETAF a Menon operator? It is conjectured, but not proven, that if A is a nonnegative square matrix satisfying certain criterion, and each of E and F is a nonnegative matrix such that ETAF is a Menon operator, then each of E and F is the product of a diagonal matrix with positive diagonal and a permutatibn matrix. This conjecture is supported by examples, and also by theorems which show that if A is doubly stochastic and ETA = TAE then either there is a number r such that rE is doubly stochastic or there is a permutation matrix P such that PtEP can be partitijned into a certain block form. A condition is defined on a doubly stochastic matrix which implies that ETA=TAE if and only if there is a number r such that rE is a permutation matrix.