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Journal ArticleDOI

Some New Inequalities via Berezin Numbers

TL;DR: In this article , the Berezin transform and the radius of an operator on the reproducing kernel Hilbert space are defined, and several sharp inequalities are studied. But they do not consider the case where the operator is a sum of two operators.
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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Journal ArticleDOI
TL;DR: The α-Berezin norm as discussed by the authors is a new norm on the space of all bounded linear operators defined on a reproducing kernel Hilbert space, which generalizes the Berezin radius.
Abstract: Abstract In this paper, we provide a new norm(α-Berezin norm) on the space of all bounded linear operators defined on a reproducing kernel Hilbert space, which generalizes the Berezin radius and the Berezin norm. We study the basic properties of the α-Berezin norm and develop various inequalities involving the α-Berezin norm. By using the inequalities we obtain various bounds for the Berezin radius of bounded linear operators, which improve on the earlier bounds. Further, we obtain a Berezin radius inequality for the sum of the product of operators, from which we derive new Berezin radius bounds.

1 citations

Journal ArticleDOI
TL;DR: For a bounded linear operator on a functional Hilbert space with normalized reproducing kernel, the Berezin symbol and Berezin number of operators on functional Hilbert spaces were defined in this article .
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
Journal ArticleDOI
13 Jun 2023
TL;DR: In this article , the authors used the Alughte transform and the generalized Alughtte transform to develop the Berezin radius inequality for Hilbert space operators, which is a special case of the BIR inequality.
Abstract: – In functional analysis, linear operators induced by functions are frequently encountered; thesecontain Hankel operators, constitution operators, and Toeplitz operators. The symbol of the resultantoperator is another name for the inciting function. In many instances, a linear operator on a Hilbert spaceℋ results in a function on a subset of a topological space. As a result, we regularly investigate operatorsinduced by functions, and we may also investigate functions induced by operators. The Berezin sign is awonderful representation of an operator-function relationship. F. Berezin proposed the Berezin switch in[8], and it has proven to be a vital tool in operator theory given that it utilizes many essential aspects ofsignificant operators. Many mathematicians and physicists are fascinated by the Berezin symbol of anoperator defined on the functional Hilbert space. The Berezin radius inequality has been extensively studiedin this situation by a number of mathematicians. In this paper, we use the Alughte transform and thegeneralized Alughte transform to develop Berezin radius inequalities for Hilbert space operators. Weadditionally offer fresh Berezin radius inequality results. Huban et al. [15] and Başaran et al. [6] supply theBerezin radius inequality.
Journal ArticleDOI
TL;DR: In this article, El-Haddad and Kittaneh (2010) show that the Hilbert uzayı üzerinde tanımlanan sınırlı lineer operatörlerin Berezin normu ve Berezin sayısı için yeni eşitsizlikler sunulmuştur.
Abstract: İşlevsel Hilbert uzayları, istatistik, yaklaşım teorisi, grup temsili teorisi, vb. dahil olmak üzere birçok alanda ortaya çıkar. İşlevsel Hilbert uzay sayesinde tanımlanan Berezin dönüşümü ise, düzgün fonksiyonları analitik fonksiyonların Hilbert uzayları üzerindeki operatörlerle ilişkilerini inceler. Berezin yarıçapını ve Berezin normunu karakterize etmek için bazı çalışmalarda birçok eşitsizlik ve bunların özellikleri vardır. Bu çalışmada fonksiyonel bir Hilbert uzayı üzerinde tanımlanan sınırlı lineer operatörlerin Berezin normu ve Berezin sayısı için yeni eşitsizlikler sunulmuştur. Bu makalenin benzersizliği veya yeniliği, iki operatör için yeni Berezin sayıları tahminlerinden oluşmaktadır. Bu tahminler, diğer benzer makaleler tarafından elde edilen Berezin sayılarının üst sınırlarını iyileştirmiştir. Daha sonra El-Haddad and Kittaneh ([10]) tarafından verilen eşitsizlik genelleştirilmiş ve iyileştirilmiştir. Bu çalışmada fikir ve sunulan metodolojiler, bu alanda gelecekteki araştırmalar için bir başlangıç noktası olarak hizmet edebilir.
Journal ArticleDOI
TL;DR: The Berezin transform and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set of sets of variables with the same re-reproducing kernel are defined in this article .
Abstract: The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $\Omega$ with the reproducing kernel $k_{\lambda}$ are defined, respectively, by \[ \widetilde{A}(\lambda)=\left\langle {A}\widehat{{k}}_{\lambda},\widehat{{k}% }_{\lambda}\right\rangle ,\ \lambda\in\Omega\text{ and }\mathrm{ber}% (A):=\sup_{\lambda\in\Omega}\left\vert \widetilde{A}{(\lambda)}\right\vert . \] We study some new inequalities by using this bounded function $\widetilde{A}, $ involving refinements of some Berezin radius inequalities for operators acting on the reproducing kernel Hilbert space.
References
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Book
01 Jan 1967

2,269 citations

Journal ArticleDOI
TL;DR: In this paper, the authors give refinements of the classical Young inequality for positive real numbers and use these refinements to establish improved Young and Heinz inequalities for matrices, which are used in this paper.

230 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that if A is a bounded linear operator on a complex Hilbert space, then w(A) ≤ 2 (A + A 2 1/2 ) where A is the usual operator norm.
Abstract: It is shown that if A is a bounded linear operator on a complex Hilbert space, then w(A) ≤ 2 (‖A‖+ ‖A2‖1/2), where w(A) and ‖A‖ are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.

212 citations