The weighted Davis-Wielandt Berezin number for reproducing kernel Hilbert space operators

A functional Hilbert space is the Hilbert space of complex-valued functions on some set Theta subset of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Theta \subseteq \mathcal {C}$\end{document} that the evaluation functionals phi lambda(f)=f(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi _{\lambda}\left ( f\right ) =f\left ( \lambda \right ) $\end{document}, lambda is an element of Theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \in \Theta $\end{document} are continuous on H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}$\end{document}. Then, by the Riesz representation theorem, there is a unique element k lambda is an element of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k_{\lambda}\in \mathcal {H}$\end{document} such that f(lambda)=< f,k lambda >\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\left ( \lambda \right ) =\left \langle f,k_{\lambda}\right \rangle $\end{document} for all f is an element of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in \mathcal {H}$\end{document} and every lambda is an element of Theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \in \Theta $\end{document}. The function k on Theta x Theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Theta \times \Theta $\end{document} defined by k(z,lambda)=k lambda(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\left ( z,\lambda \right ) =k_{\lambda}\left ( z\right ) $\end{document} is called the reproducing kernel of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}$\end{document}. In this study, we defined the weighted Davis-Wielandt Berezin number, and then we obtained some related inequalities. It is shown, among other inequalities, that if X is an element of L(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X\in{\mathcal {L}}({\mathcal {H}})$\end{document} and nu is an element of[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nu \in [0,1]$\end{document}, then 12(ber2(X nu+|X nu|2)+cber2(X nu-|X nu|2))<= dwber nu 2(X)<= 12(ber2(X nu+|X nu|2)+ber2(X nu-|X nu|2)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \frac{1}{2}\Big(\textbf{ber}<^>{2} (X_{\nu}+\vert X_{\nu}\vert <^>{2})&+c_{ \textbf{ber}}<^>{2}(X_{\nu}-\vert X_{\nu}\vert <^>{2})\Big) \\ &\leq dw_{\textbf{ber}_{\nu}}<^>{2}(X) \\ &\leq \frac{1}{2}\left (\textbf{ber}<^>{2} (X_{\nu}+\vert X_{\nu}\vert <^>{2})+ \textbf{ber}<^>{2}(X_{\nu}-\vert X_{\nu}\vert <^>{2})\right ), \end{aligned}$$ \end{document} where X nu=(1-2 nu)X & lowast;+X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X_{\nu}= (1-2\nu )X<^>{*}+X$\end{document}. Some bounds for the weighted Davis-Wielandt Berezin number are also established.