Some refinements of Berezin number inequalities via convex functions

Authors : Suna SALTAN, Nazlı BASKAN

Pages : 32-42

Doi:10.31801/cfsuasmas.1089790

View : 15 | Download : 10

Publication Date : 2023-03-30

Article Type : Research Paper

Abstract :The Berezin transform $\\widetilde{A}$ and the Berezin number of an operator $A$ on the reproducing kernel Hilbert space over some set $\\Omega$ with normalized reproducing kernel $\\widehat{k}_{\\lambda}$ are defined, respectively, by $\\widetilde{A}insert ignore into journalissuearticles values(\\lambda);=\\left\\langle {A}\\widehat{k}_{\\lambda },\\widehat{k}_{\\lambda}\\right\\rangle ,\\ \\lambda\\in\\Omega$ and $\\mathrm{ber}% insert ignore into journalissuearticles values(A);:=\\sup_{\\lambda\\in\\Omega}\\left\\vert \\widetilde{A}{insert ignore into journalissuearticles values(\\lambda);}\\right\\vert .$ A straightforward comparison between these characteristics yields the inequalities $\\mathrm{ber}\\leftinsert ignore into journalissuearticles values( A\\right); \\leq\\frac{1}{2}\\leftinsert ignore into journalissuearticles values( \\left\\Vert A\\right\\Vert _{\\mathrm{ber}}+\\left\\Vert A^{2}\\right\\Vert _{\\mathrm{ber}}% ^{1/2}\\right); $. In this paper, we study further inequalities relating them. Namely, we obtained some refinement of Berezin number inequalities involving convex functions. In particular, for $A\\in\\mathcal{B}\\leftinsert ignore into journalissuearticles values( \\mathcal{H}% \\right); $ and $r\\geq1$ we show that \\[ \\mathrm{ber}^{2r}\\leftinsert ignore into journalissuearticles values( A\\right); \\leq\\frac{1}{4}\\leftinsert ignore into journalissuearticles values( \\left\\Vert A^{\\ast }A+AA^{\\ast}\\right\\Vert _{\\mathrm{ber}}^{r}+\\left\\Vert A^{\\ast}A-AA^{\\ast }\\right\\Vert _{\\mathrm{ber}}^{r}\\right); +\\frac{1}{2}\\mathrm{ber}^{r}\\leftinsert ignore into journalissuearticles values( A^{2}\\right); . \\]
Keywords : Berezin number, Berezin transform, Cauchy Schwarz inequality, Young inequality, reproducing kernel Hilbert space