We also give some inequalities regarding the Berezin transforms of sum of two operators.
📌 3 Key Takeaways
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Objective & Hypothesis:The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator $T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $ over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined, respectively, by \[ \widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta }\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}% }\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}(T):=\sup_{\eta\in Q}\left\vert \widetilde{T}{(\eta)}\right\vert . \] We study several sharp inequalities by using this bounded function $\widetilde{T},$ involving powers of the Berezin radius and the Berezin norms of reproducing kernel Hilbert space operators.
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Key Findings & Data:The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator $T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $ over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined, respectively, by \[ \widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta }\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}% }\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}(T):=\sup_{\eta\in Q}\left\vert \widetilde{T}{(\eta)}\right\vert . \] We study several sharp inequalities by using this bounded function $\widetilde{T},$ involving powers of the Berezin radius and the Berezin norms of reproducing kernel Hilbert space operators.
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Conclusion & Impact:We also give some inequalities regarding the Berezin transforms of sum of two operators.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Reproducing kernel Hilbert space · Berezin transform · Berezin radius
Abstract
The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator $T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $ over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined, respectively, by \[ \widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta }\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}% }\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}(T):=\sup_{\eta\in Q}\left\vert \widetilde{T}{(\eta)}\right\vert . \] We study several sharp inequalities by using this bounded function $\widetilde{T},$ involving powers of the Berezin radius and the Berezin norms of reproducing kernel Hilbert space operators. We also give some inequalities regarding the Berezin transforms of sum of two operators.
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