Inequalities for the Berezin number of operators and related questions

Garayev, Mübariz T.; Alomari, Mohammad W.

doi:10.1007/s11785-021-01078-7

Cited by 19 publications

(13 citation statements)

References 29 publications

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“…Recall that the Berezin symbol A of an operator on the reproducing kernel Hilbert space H = H (Ω) with reproducing kernel k H,λ is defined by Note that C− unitary operator is a generalization of unitary (C = I ) and essentially unitary ( C = I + K, where K is compact ) operators on a Hilbert space. So, the main result of this section (Theorem 4.1 ) improves some results of the works [9,11,14,15].…”

Section: On the C− Unitarity Of C− Invertible Operatorssupporting

confidence: 72%

Some applications of Berezin θ-sequences and Berezin symbols

Altwaijry

Garayev

Baazeem

2021

Filomat

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We consider the so-called Berezin ?-sequence, where ? is an operator valued inner function, for operators on the vector valued Hardy space H2E (D), and study the invertibility of some operators on the model space K? = H2E ? ?H2E via Berezin ?-sequence. By applying of Berezin symbols technique the Toeplitz corona problem in the Bergman space L2a(D) is studied. Moreover, C-invertibility and C-unitarity of operators are also defined and studied.

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Section: On the C− Unitarity Of C− Invertible Operatorssupporting

confidence: 72%

Some applications of Berezin θ-sequences and Berezin symbols

Altwaijry

Garayev

Baazeem

2021

Filomat

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“…Berezin set and Berezin radius of operators are new numerical properties of RKHS operators presented by Karaev in [21]. See [5,12,23] for the fundamental features and information about these new categories.…”

Section: Ber (A) := Rangementioning

confidence: 99%

Improvements of some Berezin radius inequalities

Gürdal

Alomari

2022

Constructive Mathematical Analysis

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The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $Q$ with normalized reproducing kernel $k_{\eta}:=\dfrac{K_{\eta}}{\left\Vert K_{\eta}\right\Vert}$ are defined, respectively, by $\widetilde{A}(\eta)=\left\langle {A}k_{\eta},k_{\eta}\right\rangle$, $\eta\in Q$ and $\mathrm{ber} (A):=\sup_{\eta\in Q}\left\vert \widetilde{A}{(\eta)}\right\vert$. A simple comparison of these properties produces the inequalities $\dfrac{1}{4}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert \leq\mathrm{ber}^{2}\left( A\right) \leq\dfrac{1}{2}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert $. In this research, we investigate other inequalities that are related to them. In particular, for $A\in\mathcal{L}\left( \mathcal{H}\left(Q\right) \right) $ we prove that$\mathrm{ber}^{2}\left( A\right) \leq\dfrac{1}{2}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert _{\mathrm{ber}}-\dfrac{1}{4}\inf_{\eta\in Q}\left(\left( \widetilde{\left\vert A\right\vert }\left( \eta\right)\right)-\left( \widetilde{\left\vert A^{\ast}\right\vert }\left( \eta\right)\right) \right) ^{2}.$

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“…Some other quite interesting results involving applications of the Berezin transform include the characterization of invertible operators which are unitary [32], characterizations of Schatten-von Neumann class membership [11,26,29], Beurling-Arveson type theorems for some RKHS's [28], a characterization of skew-symmetric operators [3], and the characterization of truncated Toeplitz operators with bounded symbols, along with descriptions of invariant subspaces of isometric composition operators [18]. See also [30], which will be discussed in Section 5.…”

Section: Background and Motivationmentioning

confidence: 99%

Convexity of the Berezin Range

Cowen,

Felder

2021

Preprint

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This paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B(T ) := { T kx, kx H : x ∈ X}, where kx is the normalized reproducing kernel for H at x ∈ X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.

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Inequalities for the Berezin number of operators and related questions

Cited by 19 publications

References 29 publications

Some applications of Berezin θ-sequences and Berezin symbols

Some applications of Berezin θ-sequences and Berezin symbols

Improvements of some Berezin radius inequalities

Convexity of the Berezin Range

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