Author

# Mualla Birgül Huban

Bio: Mualla Birgül Huban is an academic researcher from Süleyman Demirel University. The author has contributed to research in topics: Mathematics & Hilbert space. The author has an hindex of 2, co-authored 2 publications receiving 9 citations.

##### Papers

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TL;DR: In this article, the authors generalized the Wijsman lacunary invariant statistical convergence of closed sets in metric space by introducing a new notation for the sets of triple sequences.

Abstract: In this paper, we generalized the Wijsman lacunary invariant statistical convergence of closed sets in metric space by introducing the Wijsman lacunary invariant statistical φ̃ convergence for the sets of triple sequences. We introduce the concepts of Wijsman invariant φ̃ -convergence, Wijsman invariant statistical φ̃ -convergence, Wijsman lacunary invariant φ̃ -convergence, Wijsman lacunary invariant statistical φ̃ -convergence for the sets of triple sequences. In addition, we investigate existence of some relations among these new notations for the sets of triple sequences.

8 citations

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6 citations

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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.

6 citations

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TL;DR: In this paper , the authors studied the inner product inequalities of the Berezin number of an operator A on the reproducing kernel Hilbert space H (?) over some set H (?) with the Reproducing kernel k? and established some inequalities involving the inner products of these inequalities.

Abstract: 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.

4 citations

##### Cited by

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01 Jan 2016

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78 citations

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TL;DR: In this article, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated, namely, invariant subspaces of weighted composition.

Abstract: In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition ope...

13 citations

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01 Jun 2020TL;DR: In this article, the authors used Kantorovich and KNF type inequalities in order to prove new Berezin number inequalities for powers of f (A), where A is a self-adjoint operator on the Hardy space H 2(D) and f is a positive continuous function.

Abstract: In this article, we use Kantorovich and Kantorovich type inequalities in order to prove some new Berezin number inequalities. Also, by using a refinement of the classical Schwarz inequality, we prove Berezin number inequalities for powers of f (A), where A is self-adjoint operator on the Hardy space H 2(D) and f is a positive continuous function. Some related questions are also discussed.

11 citations

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TL;DR: In this article, the authors obtained some Berezin number inequalities based on the definition of the Berezin symbol, and showed that if A,B are positive definite operators in B(H), and A#B is the geometric mean of them, then ber2(A#B)? ber (A2+B2/2)- 1/2 inf????(?k); where?(k?)
=?(A-B)?k?,?k??2, and?k? is the normalized reproducing kernel of the space H for?

Abstract: In this paper, we obtain some Berezin number inequalities based on the
definition of Berezin symbol. Among other inequalities, we show that if A,B
be positive definite operators in B(H), and A#B is the geometric mean of
them, then ber2(A#B) ? ber (A2+B2/2)- 1/2 inf ????(?k); where ?(?k?)
= ?(A-B)?k?,?k??2, and ?k? is the normalized reproducing kernel of the
space H for ? belong to some set.

9 citations

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TL;DR: In this paper, the uncertainty principle for the Berezin set and the number of operators of a bounded linear operator was studied and the spectrum and compactness of functions of model operators in terms of Berezin symbols were discussed.

Abstract: For a bounded linear operator, acting in the reproducing kernel Hilbert space $${\mathcal {H}}={\mathcal {H}}\left( \Omega \right) $$
over some set $$\Omega $$
, its Berezin symbol (or Berezin transform)
$$\widetilde{\text { }A}$$
is defined by $$\begin{aligned} {\widetilde{A}}\left( \lambda \right) :=\left\langle A{\widehat{k}}_{\lambda },{\widehat{k}}_{\lambda }\right\rangle ,\text { }\lambda \in \Omega , \end{aligned}$$
which is a bounded complex-valued function on $$\Omega ;$$
here $${\widehat{k}}_{\lambda }:=\frac{{\widehat{k}}_{\lambda }}{\left\| k_{\lambda }\right\| _{{\mathcal {H}}}}$$
is the normalized reproducing kernel of $${\mathcal {H}}$$
. The Berezin set and the Berezin number of an operator A are defined respectively by $$\begin{aligned} \mathrm {Ber}\left( A\right) :=\mathrm {Range}\left( {\widetilde{A}}\right) =\left\{ {\widetilde{A}}\left( \lambda \right) :\lambda \in \Omega \right\} \end{aligned}$$
and $$\begin{aligned} \mathrm {ber}\left( A\right) :=\sup \left\{ \left| \gamma \right| :\gamma \in \mathrm {Ber}\left( A\right) \right\} =\sup _{\lambda \in \Omega }\left| {\widetilde{A}}\left( \lambda \right) \right| . \end{aligned}$$
Since $$\mathrm {Ber}\left( A\right) \subset W\left( A\right) $$
(numerical range) and $$\mathrm {ber}\left( A\right) \le w\left( A\right) $$
(numerical radius), it is natural to investigate these new numerical quantities of operators and to get some similar results as for numerical range and numerical radius. In this paper, we prove many different type inequalities, including power inequality $$\mathrm {ber}\left( A^{n}\right) \le \mathrm {ber}\left( A\right) ^{n}$$
for the Berezin number of operators. We also study the uncertainty principle for Berezin symbols and we describe spectrum and compactness of functions of model operator in terms of Berezin symbols. Some related problems for de Branges-Rovnyak space operators are also discussed.

6 citations