Three operators' product norm and numerical radius inequalities in terms of the Moore–Penrose inverse

In this paper, we developed new norm and numerical radius inequalities for products of three bounded linear operators on a complex Hilbert space by using the Moore–Penrose inverse associated with closed-range operators. Starting from Schwarz-type vector inequalities, we derived a family of estimates that connect the norm and numerical radius of a product with positive operator expressions involving the factors and their adjoints. The results were then specialized to closed-range operators and, in particular, to inequalities involving a single operator and its Moore–Penrose inverse. Several consequences were obtained for powers of the numerical radius, including refinements and complementary bounds related to recent estimates in the literature. We also discussed the case of invertible operators, where the Moore–Penrose inverse reduces to the usual inverse, and obtained corresponding bounds that can be expressed using weighted combinations of positive operators. A numerical example for a two-by-two invertible matrix was included to illustrate the validity of one of the derived inequalities. The obtained results provide new tools for estimating the numerical radius and the operator norm in settings where Moore–Penrose inverse techniques and closed-range assumptions naturally arise. They further indicate how inequalities for products of operators can be transferred into practical estimates for individual operators through suitable substitutions and operator decompositions.