Inequalities for the Berezin number of operators and related questions

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
M. T. Garayev ◽  
M. W. Alomari
Keyword(s):  
Berezin Number

Further refinements of the Berezin number inequalities on operators

2021 ◽  
pp. 1-10
Author(s):  
Ulaş Yamancı ◽  
İsmail Murat Karlı
Keyword(s):  
Berezin Number

Berezin Number, Grüss-Type Inequalities and Their Applications

2019 ◽  
Vol 43 (3) ◽  
pp. 2287-2296 ◽  
Author(s):  
Ulaş Yamancı ◽  
Remziye Tunç ◽  
Mehmet Gürdal
Keyword(s):  
Berezin Number ◽  
Grüss Type Inequalities

scholarly journals Cebysev’s type inequalities and power inequalities for the Berezin number of operators

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2307-2316
Author(s):  
Mubariz Garayev ◽  
Ulaş Yamancı
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Reproducing Kernel ◽  
Type Inequality ◽  
Reproducing Kernel Hilbert Spaces ◽  
Selfadjoint Operators ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

We give operator analogues of some classical inequalities, including Cebysev type inequality for synchronous and convex functions of selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs). We obtain some Berezin number inequalities for the product of operators. Also, we prove the Berezin number inequality for the commutator of two operators.

scholarly journals Berezin number inequalities for operators

2019 ◽  
Vol 6 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Mojtaba Bakherad ◽  
Mubariz T. Garayev
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Berezin Transform ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Multiplicative Inequality ◽  
Berezin Number ◽  
Inequalities For Operators

Abstract The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} _\lambda } = {{{k_\lambda }} \over {\left\| {{k_\lambda }} \right\|}}$ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ${\bf{ber}}{\rm{(}}A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} }_\lambda },{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} }_\lambda }} \right\rangle } \right|$ . In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then $${\bf{ber}}(AX \pm XA) \leqslant {\bf{be}}{{\bf{r}}^{{1 \over 2}}}\left( {A*A + AA*} \right){\bf{be}}{{\bf{r}}^{{1 \over 2}}}\left( {X*X + XX*} \right)$$ and $${\bf{be}}{{\bf{r}}^2}({A^*}XB) \leqslant {\left\| X \right\|^2}{\bf{ber}}({A^*}A){\bf{ber}}({B^*}B).$$ We also prove the multiplicative inequality $${\bf{ber}}(AB){\bf{ber}}(A){\bf{ber}}(B)$$

scholarly journals New estimations for the Berezin number inequality

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mojtaba Bakherad ◽  
Ulas Yamancı
Keyword(s):  
Berezin Number

Berezin number and numerical radius inequalities for operators on Hilbert spaces

2020 ◽  
Vol 5 (3) ◽  
pp. 714-727
Author(s):  
Satyajit Sahoo ◽  
Namita Das ◽  
Debasisha Mishra
Keyword(s):  
Hilbert Spaces ◽  
Numerical Radius ◽  
Berezin Number ◽  
Inequalities For Operators

On Berezin Number Inequalities for Operator Matrices

2021 ◽  
Author(s):  
Satyajit Sahoo ◽  
Namita Das ◽  
Nirmal Chandra Rout
Keyword(s):  
Operator Matrices ◽  
Berezin Number

scholarly journals Some classical inequalities and their applications

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2165-2173
Author(s):  
Birgül Huban ◽  
Mehmet Gürdal ◽  
Havva Tilki
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Symbols Of Operators ◽  
Berezin Number ◽  
New Inequalities

In this paper, we define analogies of classical H?lder-McCarthy and Young type inequalities in terms of the Berezin symbols of operators on a reproducing kernel Hilbert space H = H (?). These inequalities are applied in proving of some new inequalities for the Berezin number of operators. We also define quasi-paranormal and absolute-k-quasi paranormal operators and study their properties by using the Berezin symbols.

scholarly journals Some extended Berezin number inequalities

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2043-2053
Author(s):  
Satyajit Sahoo ◽  
Mojtaba Bakherad
Keyword(s):  
Berezin Number

We present generalized extensions of Berezin number inequalities involving the Euclidean Berezin number and f-connection of operators.

scholarly journals Berezin number inequality for convex function in reproducing Kernel Hilbert Space

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5711-5717 ◽  
Author(s):  
Ulaş Yamancı ◽  
Mehmet Gürdal ◽  
Mubariz Garayev
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Hilbert Spaces ◽  
Convex Functions ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Hilbert Spaces ◽  
Selfadjoint Operators ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

By using Hardy-Hilbert?s inequality, some power inequalities for the Berezin number of a selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs) with applications for convex functions are given.

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