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Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2026 / Volume: 72 Issue: 2

Advanced refinements of Berezin number inequalities

Mehmet Gürdal(SÜLEYMAN DEMİREL ÜNİVERSİTESİ)0000-0003-0866-1869
Hamdullah Başaran(SÜLEYMAN DEMİREL ÜNİVERSİTESİ)0000-0002-9864-9515
Pages
—
Publication Date
2023-06-23
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In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces

📌 3 Key Takeaways

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Objective & Hypothesis:For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{\eta},\widehat{k}_{\eta}\right\rangle _{\mathcal{H}}$ and $\mathrm{ber}(A):=\sup_{\eta\in\Omega}\left\vert \widetilde{A}{(\eta)}\right\vert .$ A simple comparison of these properties produces the inequality $\mathrm{ber}% \left( A\right) \leq\frac{1}{2}\left( \left\Vert A\right\Vert_{\mathrm{ber}}+\left\Vert A^{2}\right\Vert _{\mathrm{ber}}^{1/2}\right) $ (see [17]).
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Key Findings & Data:For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{\eta},\widehat{k}_{\eta}\right\rangle _{\mathcal{H}}$ and $\mathrm{ber}(A):=\sup_{\eta\in\Omega}\left\vert \widetilde{A}{(\eta)}\right\vert .$ A simple comparison of these properties produces the inequality $\mathrm{ber}% \left( A\right) \leq\frac{1}{2}\left( \left\Vert A\right\Vert_{\mathrm{ber}}+\left\Vert A^{2}\right\Vert _{\mathrm{ber}}^{1/2}\right) $ (see [17]).
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Conclusion & Impact:In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Berezin symbol · Berezin number · functional Hilbert space

Abstract

For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{\eta},\widehat{k}_{\eta}\right\rangle _{\mathcal{H}}$ and $\mathrm{ber}(A):=\sup_{\eta\in\Omega}\left\vert \widetilde{A}{(\eta)}\right\vert .$ A simple comparison of these properties produces the inequality $\mathrm{ber}% \left( A\right) \leq\frac{1}{2}\left( \left\Vert A\right\Vert_{\mathrm{ber}}+\left\Vert A^{2}\right\Vert _{\mathrm{ber}}^{1/2}\right) $ (see [17]). In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces

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Cite This Article

Gürdal, M., Başaran, H. (2023). Advanced refinements of Berezin number inequalities. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 72 Issue: 2. https://doi.org/10.31801/cfsuasmas.1160606

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