Some properties of the q -numerical radius
Hranislav Stanković, Mihailo Krstić, Ivan Damnjanović · Linear and Multilinear Algebra · 2024
In the given study, we consider the q-numerical radius ωq(⋅) of operator matrices defined on a direct sum of Hilbert spaces and investigate the various inequalities involving these values. We also prove that supn∈Nωq(An)⩽ωq(⨁n=1+∞An)⩽|q|+21−|q|2|q|supn∈Nωq(An) for all the q∈C∖{0},|q|⩽1, thereby extending the well known equality regarding the numerical radii that occurs when we plug in q = 1. Subsequently, we give explicit formulae for computing ωq(⋅) for some special cases of operator matrices. Finally, we establish some analytical properties of ωq(⋅) regarded as a function in q.