Further generalized numerical radius inequalities for Hilbert space operators

Abstract

This paper introduces a novel and versatile framework for numerical radius inequalities within complex Hilbert spaces, building upon the generalized real and imaginary parts of an operator defined by Kittaneh and Stojiljković [Kittaneh F, Stojiljković V. New generalized numerical radius inequalities for Hilbert space operators. J Inequal Appl. 2026: 25. doi:10.1186/s13660-026-03438-3]. We define a new generalized numerical radius, 𝑤Reℎ,𝑔⁡(𝐴), and show its properties as a norm on the 𝐶∗-algebra of bounded linear operators, ℬ⁡(ℋ), under specified conditions. The proposed framework encompasses existing definitions and generalizations, yielding new identities and refined bounds for 𝑤Reℎ,𝑔⁡(𝐴). The adaptability of 𝑤Re ℎ,𝑔⁡(𝐴) through the functions h and g allows it to reduce to well-known inequalities already established in the literature, including those by Sheikhhosseini et al. [Sheikhhosseini A, Khosravi M, Sababheh M. The weighted numerical radius. Ann Funct Anal. 2022;13:3. doi:10.1007/s43034-021-00148-3] and Kittaneh [Kittaneh F. Numerical radius inequalities for Hilbert space operators. Studia Math. 2005;168:73–80. doi: 10.4064/sm168-1-5]. The work further explores various inequalities, including those involving powers of operators and operator matrices, providing extensions and refinements to previous results in the field.