Some Berezin number inequalities for operator matrices

Bakherad, Mojtaba

Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 997-1009.

Résumé

The Berezin symbol $\tilde {A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathcal H}={\mathcal H}(\Omega )$ over some (nonempty) set is defined by $\tilde {A}(\lambda )=\langle A\hat {k}_{\lambda },\hat {k}_{\lambda }\rangle ,$ $\lambda \in \Omega $, where $\hat {k}_{\lambda }={{k}_{\lambda }}/{\|{k}_{\lambda }\|}$ is the normalized reproducing kernel of ${\mathcal H}$. The Berezin number of the operator $A$ is defined by ${\bf ber}(A)=\sup _{\lambda \in \Omega }|\tilde {A}(\lambda )|=\sup _{\lambda \in \Omega }|\langle A\hat {k}_{\lambda },\hat {k}_{\lambda }\rangle |$. Moreover, ${\bf ber}(A)\leq w(A)$ (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if ${\bf T}=\left [\smallmatrix A\\ C \endmatrix \right ]\in {\mathbb B}({\mathcal H(\Omega _1)}\oplus {\mathcal H(\Omega _2)})$, then $$ {\bf ber}({\bf T}) \leq \frac {1}{2}({\bf ber}(A)+{\bf ber}(D))+\frac {1}{2}\sqrt {({\bf ber}(A)- {\bf ber}(D))^2+(\|B\|+\|C\|)^2}. $$

Détail

DOI : 10.21136/CMJ.2018.0048-17

Classification : 15A60, 30E20, 47A12, 47A30, 47B15, 47B20
Keywords: reproducing kernel; Berezin number; numerical radius; operator matrix