Some Bounds for the Generalized Spherical Numerical Radius of Operator Pairs with Applications

Najla Altwaijry, Silvestru Sever Dragomir, Kais Feki and Shigeru Furuichi ()
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Najla Altwaijry: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Silvestru Sever Dragomir: Applied Mathematics Research Group, ISILC, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia
Kais Feki: Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Faculty of Sciences of Sfax, University of Sfax, Sfax 3018, Tunisia
Shigeru Furuichi: Department of Information Science, College of Humanities and Sciences, Nihon University, Tokyo 156-8550, Japan

Mathematics, 2025, vol. 13, issue 7, 1-27

Abstract: This paper investigates a generalization of the spherical numerical radius for a pair ( B , C ) of bounded linear operators on a complex Hilbert space H . The generalized spherical numerical radius is defined as w p ( B , C ) : = sup x ∈ H , ∥ x ∥ = 1 | ⟨ B x , x ⟩ | p + | ⟨ C x , x ⟩ | p 1 p , p ≥ 1 . We derive lower bounds for w p 2 ( B , C ) involving combinations of B and C , where p > 1 . Additionally, we establish upper bounds in terms of operator norms. Applications include the cases where ( B , C ) = ( A , A * ) , with A * denoting the adjoint of a bounded linear operator A , and ( B , C ) = ( R ( A ) , I ( A ) ) , representing the real and imaginary parts of A , respectively. We also explore applications to the so-called Davis–Wielandt p -radius for p ≥ 1 , which serves as a natural generalization of the classical Davis–Wielandt radius for Hilbert-space operators.

Keywords: generalized spherical numerical radius; bounded linear operators; operator pairs; Davis–Wielandt radius; operator norms; Hilbert space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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