Generalized numerical radius inequalities for certain operator matrices

TL;DR

This work develops sharp $q$-numerical radius bounds for several structured $n\times n$ operator matrices, including tridiagonal, anti-tridiagonal, circulant, skew circulant, and their imaginary variants. By employing unitary similarity transforms (sine-based and root-of-unity-based decompositions) and leveraging key $q$-numerical radius properties, the authors express $w_q$ of complex matrices in terms of simpler scalar mixtures and derive explicit upper and lower bounds that generalize classical numerical radius results to the $q$-setting. The paper provides a unifying framework with several special cases and corollaries, along with illustrative examples and a numerical algorithm for obtaining $W_q$-bound estimates. These results advance the analysis of operator matrices in functional analysis and have potential implications for stability and spectral estimates in applications involving structured matrices.

Abstract

In this article, a series of new inequalities involving the $q$-numerical radius for $n\times n$ tridiagonal, and anti-tridiagonal operator matrices has been established. These inequalities serve to establish both lower and upper bounds for the $q$-numerical radius of operator matrices. Additionally, we developed $q$-numerical radius inequalities for $n\times n$ circulant, skew circulant, imaginary circulant, imaginary skew circulant operator matrices. Important examples have been used to illustrate the developed inequalities. In this regard, analytical expressions and a numerical algorithm have also been employed to obtain the $q$-numerical radii. We also provide a concluding section, which may lead to several new problems in this area.

Figures (5)

• Figure 1: Graphical representation a tridiagonal and anti-tridiagonal matrix.
• Figure 2: Graphical representation of a circulant matrix.
• Figure 3: The boundary of the $q$-numerical range of matrix $\mathbb{A}$ for $q = 0.5$ (left column). Comparison of $w_q\left(\mathbb{A}\right)$ with upper and lower bounds \ref{['Eqrem']} (right column). Here $\mathbb{A}$ is the tridiagonal matrix defined in Example \ref{['Ex_0']}.
• Figure 4: The boundary of the $q$-numerical range of matrix $\mathbb{A}$ for $q = 0.5$ (left column). Comparison of $w_q\left(\mathbb{A}\right)$ with upper and lower bounds \ref{['eqsymmetric']} (right column). Here $\mathbb{A}=\frac{1}{10}\frac{1}{24}\frac{1}{24}\frac{1}{10}$.
• Figure 5: The boundary of the $q$-numerical range of matrix $\mathbb{A}$ for $q = 0.5$ (left column). Comparison of $w_q\left(\mathbb{A}\right)$ with upper and lower bounds \ref{['eqsymmetric_2']} (right column). Here $\mathbb{A}=1111$.

Theorems & Definitions (39)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• proof
• Remark 3.1
• Example 3.2
• Remark 3.3