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<p>In this paper, we establish upper bounds for the <i>q</i>-Berezin radii of bounded linear operators acting on reproducing kernel Hilbert spaces. Specifically, we derive inequalities for the sum <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {X}+\mathcal {Y}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">X</mi> <mo>+</mo> <mi mathvariant="script">Y</mi> </mrow> </math></EquationSource> </InlineEquation> and product <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {Y}^*\mathcal {X}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">Y</mi> </mrow> <mo>∗</mo> </msup> <mi mathvariant="script">X</mi> </mrow> </math></EquationSource> </InlineEquation> of operators utilizing the Berezin norms of operator powers and integral refinements of the Cauchy-Schwarz inequality. We further provide explicit estimates for the <i>q</i>-Berezin radii of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(2\times 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> block operator matrices bounding the off-diagonal terms in relation to the diagonal entries. The obtained results recover the standard Berezin number inequalities when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(q=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and provide stronger estimates than the existing upper bounds for the general <i>q</i>-numerical radius.</p>

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Refined q-Berezin Radius Inequalities for Operators and \(2\times 2\) Block Matrices

  • 期刊论文

  • 发表日期：

    2026年4月18日

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  • Mehmet Gürdal,
  • Fuad Kittaneh,
  • Vuk Stojiljković

摘要

In this paper, we establish upper bounds for the q-Berezin radii of bounded linear operators acting on reproducing kernel Hilbert spaces. Specifically, we derive inequalities for the sum \(\mathcal {X}+\mathcal {Y}\) X + Y and product \(\mathcal {Y}^*\mathcal {X}\) Y ∗ X of operators utilizing the Berezin norms of operator powers and integral refinements of the Cauchy-Schwarz inequality. We further provide explicit estimates for the q-Berezin radii of \(2\times 2\) 2 × 2 block operator matrices bounding the off-diagonal terms in relation to the diagonal entries. The obtained results recover the standard Berezin number inequalities when \(q=1\) q = 1 and provide stronger estimates than the existing upper bounds for the general q-numerical radius.