The weighted Hilbert–Schmidt numerical radius

Abstract

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H and let N(⋅) be a norm on B(H). For every 0≤ν≤1, we introduce the w(N,ν)(A) as an extension of the classical numerical radius byw(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([0AB⁎0])−(1−2ν)2‖A−B‖22≤‖A‖2+‖B‖2. Our results extend some theorems due to F. Kittaneh et al. (2019).