Sharp lower bounds for generalized operator products

TL;DR

This paper establishes sharp lower bounds in the Loewner order for a broad class of positive semidefinite matrix products, extending previous results to non-commutative, non-associative, and infinite-dimensional settings.

Contribution

It generalizes existing lower bound results from Hadamard products to a wide class of bilinear products, including those on Hilbert spaces.

Findings

Proved sharp nonzero lower bounds for generalized matrix products.

Extended results to infinite-dimensional Hilbert space products.

Unified framework encompassing classical and new product types.

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.