Some operator inequalities associated with Kantorovich and Hölder–McCarthy inequalities and their applications

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

2023

We consider operator $V$ on the reproducing kernel Hilbert space $\mathcal{H}=\mathcal{H}(\Omega)$ over some set $\Omega$ with the reproducing kernel $K_{\mathcal{H},\lambda}(z)=K(z,\lambda)$ and define A-Davis-Wielandt-Berezin radius $\eta_{A}(V)$ by the formula $\eta_{A}(V):=sup\{\sqrt{| \langle Vk_{\mathcal{H},\lambda},k_{\mathcal{H},\lambda} \rangle_{A}|^{2}+\|Vk_{\mathcal{H},\lambda}\|_{A}^{4}}:\lambda \in \Omega\}$ and $\tilde{V}$ is the Berezin symbol of $V$ where any positive operator $A$-induces a semi-inner product on $\mathcal{H}$ is defined by $\langle x,y \rangle_{A}=\langle Ax,y \rangle$ for $x,y \in \mathcal{H}.$ We study equality of the lower bounds for A-Davis-Wielandt-Berezin radius mentioned above. We establish some lower and upper bounds for the A-Davis-Wielandt-Berezin radius of reproducing kernel Hilbert space operators. In addition, we get an upper bound for the A-Davis-Wielandt-Berezin radius of sum of two bounded linear operators.

Section: Introductionmentioning

confidence: 99%

“…The Berezin transform and Berezin radius have been studied by many mathematicians over the years (see [3,4,14,26]).…”

Section: Introductionmentioning

confidence: 99%

A-Davis-Wielandt-Berezin radius inequalities

GÜRDAL

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

2023

“…It is obvious that, the Berezin transform T is a bounded function on Q and sup η∈Q T (η) , which is called the Berezin radius (number) of operator T [22,23], does not exceed ∥T ∥, i. Berezin set and Berezin radius of operators are new numerical characteristics of operators on the RKHS which are introduced by Karaev in [22]. For the basic properties and facts on these new concepts, see [1,3,24,32].…”

Section: Introductionmentioning

confidence: 99%

Some New Inequalities via Berezin Numbers

Turkish Journal of Mathematics and Computer Science

BAŞARAN

Gürdal

2022

Self Cite

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.

“…Operatörlerin Berezin kümesi ve Berezin sayısı [18] de Karaev tarafından ÜÇHU üzerinde operatörlerin yeni nümerik karakteristiği olarak verilmiştir. Bu yeni kavramların temel özellikleri için [2,3,20,29,30] kaynaklarına bakılabilir.…”

Section: Introductionunclassified

Berezin Number Inequality for Increasing Operator Convex Function

Düzce Üniversitesi Bilim Ve Teknoloji Dergisi

BAŞARAN

Gürdal

2021

Self Cite

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.