The weighted Hilbert-Schmidt numerical radius

Let ]L$(H) be the algebra of all bounded linear operators on a Hilbert space H and let N(& BULL;) be a norm on ]L$(H). For every 0 & LE; & nu; & LE; 1, we introduce the w(N,& nu;)(A) as an extension of the classical numerical radius by w(N,& nu; )(A) := sup & theta;& ISIN;R N (& nu;ei & theta;A & PLUSMN; (1 - & nu;)e-i & theta;A* and investigate basic properties of this notion and prove inequalities involving it. In particular, when N(& BULL;) is the Hilbert-Schmidt norm & BULL;2, we present several the weighted Hilbert-Schmidt numerical radius inequalities for operator matrices. Furthermore, we give a refinement of the triangle inequality for the Hilbert-Schmidt norm as follows: & LE; A 2 & PLUSMN; B 2. Our results extend some theorems due to F. Kittaneh et al. (2019). & COPY; 2023 Elsevier Inc. All rights reserved.