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

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 wp(B,C):=supx∈H,∥x∥=1|⟨Bx,x⟩|p+|⟨Cx,x⟩|p1p, p≥1. We derive lower bounds for wp2(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.