New Approach to Generalized Berezin Norms and Rigorous Operator Bounds

Let $\left({\mathfrak{H}}_{\Theta },⟨·,·⟩\right)$ be a reproducing kernel Hilbert space over a non-empty set $\Theta$, and let $\mathrm{A}$ be a non-zero positive operator on ${\mathfrak{H}}_{\Theta }$. This operator induces a semi-inner product given by ${⟨\xi ,\eta ⟩}_{\mathrm{A}}=⟨\mathrm{A}\xi ,\eta ⟩$ for all $\xi ,\eta \in {\mathfrak{H}}_{\Theta }$, with the associated seminorm ${\parallel \xi \parallel }_{\mathrm{A}}=\sqrt{{⟨\xi ,\xi ⟩}_{\mathrm{A}}}$. The $\mathrm{A}$-normalized Berezin number and the $\mathrm{A}$-normalized Berezin norm of an $\mathrm{A}$-bounded linear operator $\mathrm{C}$ on ${\mathfrak{H}}_{\Theta }$ are defined by ${\displaystyle {\mathbf{b}}_{\mathrm{A}}\left(\mathrm{C}\right)=\underset{\gamma \in {\Theta }_{\mathrm{A}}}{sup}\left|{⟨\mathrm{C}{\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}},{\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}}⟩}_{\mathrm{A}}\right|}$ and ${\displaystyle {\parallel \mathrm{C}\parallel }_{{\mathbf{b}}_{\mathrm{A}}}=\underset{\gamma ,\delta \in {\Theta }_{\mathrm{A}}}{sup}\left|{⟨\mathrm{C}{\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}},{\widehat{\mathrm{x}}}_{\delta }^{\mathrm{A}}⟩}_{\mathrm{A}}\right|}$, where ${\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}}={\displaystyle \frac{{\mathrm{x}}_{\gamma }}{\parallel {\mathrm{x}}_{\gamma }{\parallel }_{\mathrm{A}}}}$ and ${\Theta }_{\mathrm{A}}=\{\gamma \in \Theta :\parallel {\mathrm{x}}_{\gamma }{\parallel }_{\mathrm{A}}\ne 0\}$. The primary aim of this paper is to establish new sharp bounds and inequalities involving these two quantities and related operator-theoretic notions. In doing so, we propose a novel method to address the challenges of operator bounds. Furthermore, we revisit recent results on generalized Berezin norms, in particular those of Huban’s work in 2022. We show that some of these results rely on the incorrect assumption that the $\mathrm{A}$-Berezin number coincides with the $\mathrm{A}$-Berezin norm for $\mathrm{A}$-selfadjoint operators. By providing corrected arguments and employing tools such as the $\mathrm{A}$-Cartesian decomposition and the generalized Buzano inequality, we develop a consistent and rigorous framework for the study of generalized Berezin symbols in semi-Hilbertian spaces.