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

Abstract

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.