Positive eigenvalues of generalized words in two Hermitian positive definite matrices

Abstract

We define a word in two positive definite (complex Hermitian) matrices A and B as a finite product of real powers of A and B. The question of which words have only positive eigenvalues is addressed. This question was raised some time ago in connection with a long-standing problem in theoretical physics, and it was previously approached by the authors for words in two real positive definite matrices with positive integral exponents. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do.