Abstract

In the given study, we consider the q-numerical radius ω q ( ⋅ ) of operator matrices defined on a direct sum of Hilbert spaces and investigate the various inequalities involving these values. We also prove that sup n ∈ N ω q ( A n ) ⩽ ω q ( ⨁ n = 1 + ∞ A n ) ⩽ | q | + 2 1 − | q | 2 | q | sup n ∈ N ω q ( A n ) 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 ω q ( ⋅ ) for some special cases of operator matrices. Finally, we establish some analytical properties of ω q ( ⋅ ) regarded as a function in q.

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