Sharp lower bounds for generalized operator products

Abstract

We consider general bilinear products defined by positive semidefinite matrices. Typically non-commutative, non-associative, and non-unital, these products preserve positivity and include the classical Hadamard, Kronecker, and convolutional products as special cases. We prove that every such product satisfies a sharp nonzero lower bound in the Loewner order, generalizing previous results of Vyb\'iral [Adv. Math., 2020] and Khare [Proc. Amer. Math. Soc., 2021] that were obtained in the special case of the Hadamard product. Our results naturally extend to Hilbert spaces for a family of products parametrized by positive trace-class operators, providing a lower bound in the Loewner order for such general products, including for the Hilbert tensor product.