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 $\Theta \subseteq \mathcal {C}$ that the evaluation functionals $\varphi _{\lambda}\left ( f\right ) =f\left ( \lambda \right ) $ , $\lambda \in \Theta $ are continuous on $\mathcal {H}$ . Then, by the Riesz representation theorem, there is a unique element $k_{\lambda}\in \mathcal {H}$ such that $f\left ( \lambda \right ) =\left \langle f,k_{\lambda}\right \rangle $ for all $f\in \mathcal {H}$ and every $\lambda \in \Theta $ . The function k on $\Theta \times \Theta $ defined by $k\left ( z,\lambda \right ) =k_{\lambda}\left ( z\right ) $ is called the reproducing kernel of $\mathcal {H}$ . 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 $X\in{\mathcal {L}}({\mathcal {H}})$ and $\nu \in [0,1]$ , then $$\begin{aligned} \frac{1}{2}\Big(\textbf{ber}^{2} (X_{\nu}+\vert X_{\nu}\vert ^{2})&+c_{ \textbf{ber}}^{2}(X_{\nu}-\vert X_{\nu}\vert ^{2})\Big) \\ &\leq dw_{\textbf{ber}_{\nu}}^{2}(X) \\ &\leq \frac{1}{2}\left (\textbf{ber}^{2} (X_{\nu}+\vert X_{\nu}\vert ^{2})+ \textbf{ber}^{2}(X_{\nu}-\vert X_{\nu}\vert ^{2})\right ), \end{aligned}$$ where $X_{\nu}= (1-2\nu )X^{*}+X$ . Some bounds for the weighted Davis-Wielandt Berezin number are also established.