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

Paper

Abstract

This paper investigates new properties of

-numerical ranges for compact normal operators and establishes refined upper bounds for the

-numerical radius of Hilbert space operators. We first prove that for a compact normal operator

with

, the

-numerical range

is a closed convex set containing the origin in its interior. We then explore the behavior of

-numerical ranges under complex symmetry, deriving inclusion relations between

and

for complex symmetric operators. For hyponormal operators similar to their adjoints, we provide conditions under which

is self-adjoint and

is a real interval. We also study the continuity of

-numerical ranges under norm convergence and examine the effect of the Aluthge transform on

. In the second part, we derive several new and sharp upper bounds for the

-numerical radius, incorporating the operator norm, numerical radius, transcendental radius, and the infimum of

over the unit sphere. These bounds unify and improve upon existing results in the literature, offering a comprehensive framework for estimating

-numerical radii across the entire parameter range

. Each result is illustrated with detailed examples and comparisons with prior work.