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

Abstract

Abstract A functional Hilbert space is the Hilbert space of complex-valued functions on some set Θ ⊆ C $\Theta \subseteq \mathcal {C}$ that the evaluation functionals φ λ ( f ) = f ( λ ) $\varphi _{\lambda}\left ( f\right ) =f\left ( \lambda \right ) $ , λ ∈ Θ $\lambda \in \Theta $ are continuous on H $\mathcal {H}$ . Then, by the Riesz representation theorem, there is a unique element k λ ∈ H $k_{\lambda}\in \mathcal {H}$ such that f ( λ ) = 〈 f , k λ 〉 $f\left ( \lambda \right ) =\left \langle f,k_{\lambda}\right \rangle $ for all f ∈ H $f\in \mathcal {H}$ and every λ ∈ Θ $\lambda \in \Theta $ . The function k on Θ × Θ $\Theta \times \Theta $ defined by k ( z , λ ) = k λ ( z ) $k\left ( z,\lambda \right ) =k_{\lambda}\left ( z\right ) $ is called the reproducing kernel of H $\mathcal {H}$ . 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 ∈ L ( H ) $X\in{\mathcal {L}}({\mathcal {H}})$ and ν ∈ [ 0 , 1 ] $\nu \in [0,1]$ , then 1 2 ( ber 2 ( X ν + | X ν | 2 ) + c ber 2 ( X ν − | X ν | 2 ) ) ≤ d w ber ν 2 ( X ) ≤ 1 2 ( ber 2 ( X ν + | X ν | 2 ) + ber 2 ( X ν − | X ν | 2 ) ) , $$\begin{aligned} \frac{1}{2}\Big(\textbf{ber}^{2} (X_{\nu}+\vert X_{\nu}\vert ^{2})&+c_{ \textbf{ber}}^{2}(X_{\nu}-\vert X_{\nu}\vert ^{2})\Big) \\ &\leq dw_{\textbf{ber}_{\nu}}^{2}(X) \\ &\leq \frac{1}{2}\left (\textbf{ber}^{2} (X_{\nu}+\vert X_{\nu}\vert ^{2})+ \textbf{ber}^{2}(X_{\nu}-\vert X_{\nu}\vert ^{2})\right ), \end{aligned}$$ where X ν = ( 1 − 2 ν ) X ∗ + X $X_{\nu}= (1-2\nu )X^{*}+X$ . Some bounds for the weighted Davis-Wielandt Berezin number are also established.

Keywords

• Weighted Berezin number
• Berezin set
• Berezin symbol
• Davis-Wielandt Berezin