Novel Berezin number and norm inequalities for operator sums and products

Let $({\mathcal{X}}_{\mathcal{F}},⟨\cdot ,\cdot ⟩)$ be a reproducing kernel Hilbert space over a non-empty set $\mathcal{F}$. Let ${\hat{u}}_{\lambda }$ and ${\hat{u}}_{\mu }$ denote the normalized reproducing kernels of ${\mathcal{X}}_{\mathcal{F}}$. The Berezin number and the Berezin norm of a bounded linear operator $\mathcal{B}$ acting on ${\mathcal{X}}_{\mathcal{F}}$ are, respectively, defined by

$$\mathbf{b}\mathbf{e}\mathbf{r}(\mathcal{B})=\underset{\lambda \in \mathcal{F}}{sup}|⟨\mathcal{B}{\hat{u}}_{\lambda },{\hat{u}}_{\lambda }⟩|\text{and}\Vert \mathcal{B}{\Vert }_{\mathbf{b}\mathbf{e}\mathbf{r}}=\underset{\lambda ,\mu \in \mathcal{F}}{sup}|⟨\mathcal{B}{\hat{u}}_{\lambda },{\hat{u}}_{\mu }⟩|.$$

In this work, we establish new upper bounds for these two quantities. In particular, we derive bounds for their sums and obtain novel estimates for a specific type of product, namely $\mathbf{b}\mathbf{e}\mathbf{r}({\mathcal{C}}^{\ast }\mathcal{B})$, where ${\mathcal{C}}^{\ast }$ denotes the adjoint of $\mathcal{C}$. Some of our results also involve another Berezin-type norm that is equivalent to the quantities mentioned above. Several applications and improvements of existing results in the literature are provided.