New Properties and Refined Bounds for the $q$-Numerical Range

This paper investigates new properties of $q$-numerical ranges for compact normal operators and establishes refined upper bounds for the $q$-numerical radius of Hilbert space operators. We first prove that for a compact normal operator $T$ with $0 \in W_q(T)$, the $q$-numerical range $W_q(T)$ is a closed convex set containing the origin in its interior. We then explore the behavior of $q$-numerical ranges under complex symmetry, deriving inclusion relations between $W_q(T)$ and $W_q(T^*)$ for complex symmetric operators. For hyponormal operators similar to their adjoints, we provide conditions under which $T$ is self-adjoint and $W_q(T)$ is a real interval. We also study the continuity of $q$-numerical ranges under norm convergence and examine the effect of the Aluthge transform on $W_q(T)$. In the second part, we derive several new and sharp upper bounds for the $q$-numerical radius, incorporating the operator norm, numerical radius, transcendental radius, and the infimum of $|Tx|$ over the unit sphere. These bounds unify and improve upon existing results in the literature, offering a comprehensive framework for estimating $q$-numerical radii across the entire parameter range $q \in [0,1]$. Each result is illustrated with detailed examples and comparisons with prior work.

💡 Research Summary

This preprint, titled “New Properties and Refined Bounds for the q-Numerical Range,” presents a comprehensive study on the theory of q-numerical ranges for Hilbert space operators. The work is bifurcated into two main thrusts: establishing new structural properties of q-numerical ranges for specific operator classes, and deriving a set of refined and sharp upper bounds for the q-numerical radius.

The first part delves into the theoretical foundations. A key result (Theorem 2.1) proves that for a compact normal operator T, if the origin lies within its q-numerical range W_q(T), then W_q(T) is necessarily a closed convex set containing the origin as an interior point. This extends classical numerical range properties and has implications for spectral theory. The paper then investigates operators with symmetry properties. For complex symmetric operators (satisfying T = C T* C for a conjugation C), Theorem 2.5 establishes intricate inclusion relations between W_q(T) and rotated versions of W_q(T*), leading to circular symmetry of W_0(T) for normal complex symmetric operators. Furthermore, Theorem 2.9 provides conditions under which a hyponormal operator T, similar to its adjoint via an invertible operator X, must be self-adjoint, rendering its q-numerical range a real interval. Additional analyses cover the continuity of q-numerical ranges under norm convergence and the effect of the Aluthge transform.

The second and equally significant part of the paper focuses on obtaining practical estimates. The author derives several new and sharp upper bounds for the q-numerical radius ω_q(T). These bounds innovatively combine various operator-theoretic quantities such as the operator norm ||T||, the classical numerical radius w(T), the transcendental radius m(T), and the Crawford number c(T). The derived inequalities, for instance ω_q^2(T) ≤ q^2 w^2(T) + (1-q^2 + q√(1-q^2))||T||^2 - (1-q^2)c^2(T), are shown to unify and improve upon existing results in the literature. They provide a versatile framework for estimating ω_q(T) across the entire parameter spectrum q ∈