2021/11/28 by Ali Zamani, Zamani, Ali
Computer Science · Mathematics · #47A12 #47A30 #47A63 #47B10 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.14222
openalex publication_date 2021/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbB(H) be the algebra of all bounded linear operators on a Hilbert space H and let N(⋅) be a norm on \mathbbB(H). For every 0≤ ν≤ 1, we introduce the w_(N,ν)(A) as an extension of the classical numerical radius by w_(N,ν)(A):= supθ∈ ℝ N(νeiθA + (1-ν)e-iθA^*) and investigate basic properties of this notion and prove inequalities involving it. In particular, when N(⋅) is the Hilbert--Schmidt norm ‖ ⋅ ‖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: ‖A+B‖2 ≤ √2w__(‖ ⋅ ‖2,ν)2(\beginbmatrix 0 amp; A
B^* amp; 0 \endbmatrix) - (1-2ν)2‖A-B‖22 ≤ ‖A‖2 + ‖B‖2. Our results extend some theorems due to F.~Kittaneh et al. (2019).