Author

# Hamdullah Basaran

Bio: Hamdullah Basaran is an academic researcher. The author has contributed to research in topics: Mathematics & Hilbert space. The author has an hindex of 1, co-authored 1 publications receiving 4 citations.

##### Papers

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TL;DR: In this article, the authors prove analogies of certain operator inequalities, including Hölder-McCarthy inequality, Kantorovich inequality, and Heinz-Kato inequality, for positive operators on the Hilbert space in terms of the Berezin symbols and the number of operators in the reproducing kernel Hilbert space.

Abstract: We prove analogs of certain operator inequalities, including Hölder–McCarthy inequality, Kantorovich inequality, and Heinz–Kato inequality, for positive operators on the Hilbert space in terms of the Berezin symbols and the Berezin number of operators on the reproducing kernel Hilbert space.

17 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

TL;DR: In this paper , some refinements of the Cauchy-Schwarz inequality for contraction operators on the reproducing kernel Hilbert space are given in terms of the Berezin transform.

Abstract: In this manuscript, some refinements of the Cauchy-Schwarz inequality for contraction operators on the reproducing kernel Hilbert space are given in terms of the Berezin transform. We show several additional inequalities for the Berezin norm and Berezin radius of operators using these refinements.

2 citations

TL;DR: In this paper , the Hermite-Hadamard inequalities for operators on reproducing kernel Hilbert spaces were obtained by using classical Hermite Hadamard inequality and convex functions.

Abstract: . The Berezin symbol (cid:101) A of an operator A on the reproducing kernel Hilbert space H (Ω) over some set Ω with the reproducing kernel k λ is deﬁned by ˜ A ( λ ) = (cid:68) A (cid:98) k λ , (cid:98) k λ (cid:69) , λ ∈ Ω . In this paper, we obtain some new inequalities for Berezin symbols of operators on reproducing kernel Hilbert spaces by using classical Hermite-Hadamard inequality and convex functions. Some other related questions are also discussed.

2 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 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, by using some classical operator means and classical operator inequalities, a power inequality for the Berezin number of operators was established for n-tuple operators via Berezin symbols.

Abstract: In this paper, by using some classical operator means and classical operator inequalities, we investigate Berezin number of operators. In particular, we compare the Berezin number of some operator means of two positive operators. We also use some Hardy type inequalities to obtain a power inequality for the Berezin number of an operator. Moreover, by applying some inequalities for nonnegative Hermitian forms, some vector inequalities for n-tuple operators via Berezin symbols are established.

5 citations

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TL;DR: In this paper, the Berezin number is used to define the operator geometric mean, and it is shown that if a positive operator X and Y are positive operators, then the geometric mean of the positive operator can be reduced to

Abstract: In this paper, by the definition of Berezin number, we present some inequalities involving the operator geometric mean. For instance, it is shown that if $X, Y, Z\in {\mathcal{L}}(\mathcal{H})$ such that X and Y are positive operators, then $$\begin{aligned} \operatorname{ber}^{r} \bigl( ( X\mathbin{\sharp} Y ) Z \bigr) &\leq \operatorname{ber} \biggl(\frac{ ( Z^{\star }YZ ) ^{\frac{rq}{2}}}{q}+ \frac{X^{ \frac{rp}{2}}}{p} \biggr) -\frac{1}{p}\inf_{\lambda \in \varOmega } \bigl( \bigl[ \widetilde{X} ( \lambda ) \bigr] ^{\frac{rp}{4}}- \bigl[ \widetilde{ \bigl( Z^{\star }YZ \bigr) } ( \lambda ) \bigr] ^{ \frac{rq}{4}} \bigr) ^{2}, \end{aligned}$$ in which $X\mathbin{\sharp} Y=X^{\frac{1}{2}} ( X^{-\frac{1}{2}}YX^{- \frac{1}{2}} ) ^{\frac{1}{2}}X^{\frac{1}{2}}$, $p\geq q>1$ such that $r\geq \frac{2}{q}$ and $\frac{1}{p}+\frac{1}{q}=1$.

5 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