On the estimation of the $q$-numerical radius via Orlicz functions

TL;DR

The paper develops refined bounds for the $q$-numerical radius of operators on a Hilbert space by employing Orlicz functions. It proves a central Orlicz-based two-sided inequality for $w_q(T)$, yielding concrete lower and upper bounds in terms of $\|TT^*+T^*T\|$, and extends these results to sums and products of operators. Furthermore, it introduces the notion of $q$-sectorial matrices, derives bounds for the $q$-numerical radius in terms of sectorial parameters, and provides sharper estimates for commutator and anti-commutator forms as well as generalized product expressions. Overall, the results unify and sharpen several existing bounds for the $q$-numerical radius and broaden the toolkit for operator-norm estimations in the presence of a $q$-parameter and Orlicz-analytic structure.

Abstract

This study utilizes Orlicz functions to provide refined lower and upper bounds on the q-numerical radius of an operator acting on a Hilbert space. Additionally, the concept of q-sectorial matrices is introduced and further bounds for the q-numerical radius are established. Our results unify several existing bounds for the q-numerical radius. Suitable examples are provided to supplement the estimations.

Figures (6)

• Figure 1: Comparison of $\frac{(q+2\sqrt{1-q^2})^2}{2}$ and $\frac{2-q^2+2q\sqrt{2(1-q^2)}}{2}$
• Figure 2: $q$-Sectorial matrix $A$ with $q$-sectorial index $\angle POX= \angle QOX= \alpha$.
• Figure 3: Shapes of $W_{\frac{1}{2}}(A_1)$ and $W_{\frac{1}{2}}(A_2)$.
• Figure 4: Comparison $\cos(\alpha)$ and $\frac{1}{1+\sin(\alpha)}$
• Figure 5: If $\sin(\alpha)=0.5.$

Theorems & Definitions (55)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 1.1
• Theorem 1.2
• Lemma 1.6
• Lemma 1.7
• Lemma 1.8