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<p>This paper presents several new and refined inequalities for the <i>A</i>-numerical radius of operator matrices in semi-Hilbertian spaces, where <i>A</i> is a positive operator on a complex Hilbert space. By leveraging and extending various generalized versions of the <i>A</i>-Buzano inequality, mixed Schwarz inequality, and Hölder-McCarthy inequality, we establish upper bounds for the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">A</mi> </math></EquationSource> </InlineEquation>-numerical radius of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2 \times 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> operator matrices. Our results generalize and improve upon many existing inequalities in the literature, offering sharper and more versatile bounds. The introduced inequalities are parameter-dependent, providing a flexible framework that interpolates between and unifies previous results. Specific cases and special choices of parameters are discussed, demonstrating the broad applicability and improvements achieved. An illustrative example within the Hardy space setting is provided to concretely demonstrate the application of one of our main theorems.</p>

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Advanced a-numerical radius inequalities: product operators and refined bounds

  • 期刊论文

  • 发表日期：

    2026年1月22日

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  • M. H. M Rashid

摘要

This paper presents several new and refined inequalities for the A-numerical radius of operator matrices in semi-Hilbertian spaces, where A is a positive operator on a complex Hilbert space. By leveraging and extending various generalized versions of the A-Buzano inequality, mixed Schwarz inequality, and Hölder-McCarthy inequality, we establish upper bounds for the \(\mathbb {A}\) A -numerical radius of \(2 \times 2\) 2 × 2 operator matrices. Our results generalize and improve upon many existing inequalities in the literature, offering sharper and more versatile bounds. The introduced inequalities are parameter-dependent, providing a flexible framework that interpolates between and unifies previous results. Specific cases and special choices of parameters are discussed, demonstrating the broad applicability and improvements achieved. An illustrative example within the Hardy space setting is provided to concretely demonstrate the application of one of our main theorems.