2025/12/11 by M. H. M. Rashid, Rashid, Mohammad H. M.
Mathematics · #47A12 #47A30 #47B07 #47B15 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Operator Algebras (math.OA) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2512.13719
openalex publication_date 2025/12/11 · openalex created_date 2025/12/18 · openalex updated_date 2026/07/28
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 ∈ Wq(T), the q-numerical range Wq(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 Wq(T) and Wq(T^*) for complex symmetric operators. For hyponormal operators similar to their adjoints, we provide conditions under which T is self-adjoint and Wq(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 Wq(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 ∈ [0,1]. Each result is illustrated with detailed examples and comparisons with prior work.