Improvements of some Berezin radius inequalities

Authors : Mehmet GÜRDAL, Mohammad ALOMARI

Pages : 141-153

Doi:10.33205/cma.1110550

View : 21 | Download : 11

Publication Date : 2022-09-15

Article Type : Research Paper

Abstract :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 $\widetilde{A}insert ignore into journalissuearticles values(\eta);=\left\langle {A}k_{\eta},k_{\eta}\right\rangle$, $\eta\in Q$ and $\mathrm{ber} insert ignore into journalissuearticles values(A);:=\sup_{\eta\in Q}\left\vert \widetilde{A}{insert ignore into journalissuearticles values(\eta);}\right\vert$. 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}\leftinsert ignore into journalissuearticles values( 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}\leftinsert ignore into journalissuearticles values( \mathcal{H}\leftinsert ignore into journalissuearticles values(Q\right); \right); $ we prove that $\mathrm{ber}^{2}\leftinsert ignore into journalissuearticles values( A\right); \leq\dfrac{1}{2}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert _{\mathrm{ber}}-\dfrac{1}{4}\inf_{\eta\in Q}\leftinsert ignore into journalissuearticles values(\leftinsert ignore into journalissuearticles values( \widetilde{\left\vert A\right\vert }\leftinsert ignore into journalissuearticles values( \eta\right);\right);-\leftinsert ignore into journalissuearticles values( \widetilde{\left\vert A^{\ast}\right\vert }\leftinsert ignore into journalissuearticles values( \eta\right);\right); \right); ^{2}.$
Keywords : Mixed Schwarz inequality, Berezin radius, Furuta inequality