Some Sharp bounds for the generalized Davis-Wielandt radius

This paper presents a study of the generalized Davis-Wielandt radius of Hilbert space operators. New lower bounds for the generalized Davis-Wielandt radius and numerical radius are provided. An alternative of the triangular inequality for operators is also derived.

💡 Research Summary

The paper studies a generalized Davis–Wielandt radius for bounded linear operators on a complex Hilbert space. Starting from the classical numerical radius w(T) and Davis–Wielandt radius dw(T), the authors introduce a generalized numerical radius w_N(T)=sup_{θ∈ℝ} N(Re(e^{iθ}T)), where N is an arbitrary norm on B(H). Using this, they define the generalized Davis–Wielandt radius
dw_N(T)=sup_{θ∈ℝ} √{ N^2(Re(e^{iθ}T)) + N^4(Re(e^{iθ}|T|)) }.
The paper’s main contributions are a series of sharp lower and upper bounds for dw_N(T) that improve upon earlier results, especially those in Alomari‑et‑al. (2024).

Theorem 2 provides a stronger lower bound than the previously known dw_N(T) ≥ ¼ N^2(T)+½ N^4(|T|). By evaluating the definition at θ=0 and θ=−π/2 and using the identity max{a,b}=½(a+b+|a−b|), the authors obtain
dw_N(T) ≥ ½ N^2(T) + 2 N^4(|T|) + 2 | N^2(Re T)+N^4(|T|)−N^2(Im T) |.
When N is an algebra norm, Theorem 3 yields even tighter estimates involving N(|T|^2+|T^*|^2) or N(T^2+T^{*2}). These bounds dominate the earlier inequality dw_N(T) ≥ ¼ N^2(T)+½ N^4(|T|).

Theorem 4 gives a new lower bound for the generalized numerical radius:
w_N(T) ≥ ½ max{ N(|T|^2+|T^*|^2), N(T^2+T^{*2}) },
which refines previous results on w_N.

Further, Theorem 5 and Theorem 6 introduce auxiliary quantities m₁, m₂, d₁, d₂ to combine several basic estimates into a compact expression:
dw_N(T) ≥ ¾ N^2(T) + 2 N^4(|T|) + (d₁+d₂) + 2|m₁−m₂|.
These composite bounds are shown to be sharp by explicit 2×2 matrix examples.

Because dw_N does not satisfy the triangle inequality, the authors propose an alternative sum inequality (Theorem 8):
dw_N(T+S) ≤ 2√2 dw_N(T) + dw_N(S) ≤ 2√2 (dw_N(T)+dw_N(S)).
This result improves the earlier estimate dw(T+S) ≤ √2(d w^2(T)+dw^2(S)) + 6‖|T|^4+|S|^4‖ and provides a cleaner constant.

The paper includes several illustrative examples: a non‑normal matrix T=