Operator Theory, Complex Analysis and Applications Seminar 

20/06/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building
Nelson Faustino, Universidade de Coimbra

Berezin Calculus over Weighted Bergman Spaces of Polyanalytic type

Starting from the Poincaré metric $d{s}^2=\frac{1}{2\pi i}{(1-\mid z{\mid }^2)}^{-2}d\bar{z}dz$ on the the unit disk $𝔻$, we will study the range of the Berezin transforms generated from the normalized kernel function $K_{\zeta }^n(z)=K^n(z,\zeta )K^n(\zeta ,\zeta {)}^{-\frac{1}{2}}$ regarding the weighted polyanalytic Bergman spaces $A_n^{\alpha }(𝔻)$ of order $n$. Special emphasize will be given to the invariance of the range of the Berezin transformation under the action of the Möbius transformations ${\phi }_{\zeta }(z)=\frac{z-a}{1-\bar{\zeta }z}$. Connection between Berezin calculus over weighted Bergman spaces of polyanalytic type on the disk $𝔻$ and on the upper half space ${ℂ}^{+}$ will also be discussed along the talk.