Further generalized numerical radius inequalities for Hilbert space operators

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, wh,gRe(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 wh,gRe(A). The adaptability of wh,gRe(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.