- Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics
- Volume:72 Issue:2
- Advanced refinements of Berezin number inequalities
Advanced refinements of Berezin number inequalities
Authors : Mehmet GÜRDAL, Hamdullah BAŞARAN
Pages : 386-396
Doi:10.31801/cfsuasmas.1160606
View : 69 | Download : 53
Publication Date : 2023-06-23
Article Type : Research Paper
Abstract :For a bounded linear operator $A$ on a functional Hilbert space $\\mathcal{H}\\leftinsert ignore into journalissuearticles values( \\Omega\\right); $, with normalized reproducing kernel $\\widehat {k}_{\\eta}:=\\frac{k_{\\eta}}{\\left\\Vert k_{\\eta}\\right\\Vert _{\\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\\widetilde{A}\\leftinsert ignore into journalissuearticles values( \\eta\\right); :=\\left\\langle A\\widehat{k}_{\\eta},\\widehat{k}_{\\eta}\\right\\rangle _{\\mathcal{H}}$ and $\\mathrm{ber}insert ignore into journalissuearticles values(A);:=\\sup_{\\eta\\in\\Omega}\\left\\vert \\widetilde{A}{insert ignore into journalissuearticles values(\\eta);}\\right\\vert .$ A simple comparison of these properties produces the inequality $\\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); $ insert ignore into journalissuearticles values(see [17]);. In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spacesKeywords : Berezin symbol, Berezin number, functional Hilbert space, positive operator
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