We prove analogs of certain operator inequalities, including Hölder-McCarthy inequality, Kantorovich inequality, and Heinz-Kato inequality, for positive operators on the Hilbert space in terms of the Berezin symbols and the Berezin number of operators on the reproducing kernel Hilbert space.

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.

The Berezin symbol ?A of an operator A on the reproducing kernel Hilbert space H (?) over some set ? with the reproducing kernel k? is defined by ? (?) = ?A k?/||k?||, k?/||k?||?, ? ? ?. The Berezin number of an operator A is defined by ber(A) := sup ??? |?(?)|. We study some problems of operator theory by using this bounded function ?, including treatments of inner product inequalities via convex functions for the Berezin numbers of some operators. We also establish some inequalities involving of the Berezin inequalities.

olan üretici çekirdekli ℋ(𝛺) Hilbert uzayı üzerinde 𝐴 sınırlı lineer operatör için Berezin sembolü ve Berezin sayısı sırasıyla 𝐴 ̃(𝜆) ≔ 〈𝐴𝐾 𝜆 , 𝐾 𝜆 〉 ℋ ve 𝑏𝑒𝑟(𝐴) ≔ 𝑠𝑢𝑝 𝜆∈𝛺 |𝐴 ̃(𝜆)| biçiminde tanımlanır. Bu karakteristik ifadeler kullanılarak 𝑏𝑒𝑟(𝐴) ≤ 1 √2 𝑏𝑒𝑟(|𝐴| + 𝑖|𝐴 * |) eşitsizliği elde edilmiştir. Bu çalışmamızda ise onlar arasındaki diğer eşitsizlikler ispatlanmış ve Berezin sayı eşitsizlikleri için operatör konveks fonksiyonlarının bazı uygulamaları verilmiştir.

We investigate new upper bounds for the Berezin radius and Berezin norm of $2\times2$ operator matrices using the Cauchy-Buzano inequality, and we propose a required condition for the equality case in the triangle inequalities for the Berezin norms. We also show various Berezin radius inequalities for matrices with $2\times2$ operators.