The weighted Hilbert–Schmidt numerical radius

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H and let N(⋅) norm B(H). For every 0≤ν≤1, we introduce w(N,ν)(A) as an extension classical numerical radius byw(N,ν)(A):=supθ∈R⁡N(νeiθA+(1−ν)e−iθA⁎) investigate basic properties this notion prove inequalities involving it. In particular, when is Hilbert–Schmidt ‖⋅‖2, present several weighted for operator matrices. Furthermore, give refinement triangle inequality 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).