The Bergman kernel function
The author shows that a large family of square matrix valued kernel functions defined on the unit disc \(\mathbb D\subset\mathbb C\), which were constructed in [\textit{A. Korányi} and \textit{G. Misra}, J. Funct. Anal. 254, No. 9, 2419--2436 (2008; Zbl 1171.46022)], behave like the familiar Bergman kernel function on \(\mathbb D\). The existence of a positive definite kernel function \[ B:\mathbb D\times\mathbb D\rightarrow\mathbb C^{n\times n} \] possessing the quasi-invariance property of the Bergman kernel function is proved. Hence a slight generalization of the construction used in [\textit{A. Korányi} and \textit{G. Misra}, in: Perspectives in mathematical sciences II. Pure mathematics. Papers of the conference on perspectives in mathematical sciences, Bangalore, India, February 4--8, 2008. Hackensack, NJ: World Scientific. Statistical Science and Interdisciplinary Research 8; Platinum Jubilee Series, 83--101 (2009; Zbl 1192.47021)] implies the existence of a Hilbert space \(\mathcal H\) of holomorphic functions on \(\mathbb D\) taking values in \(\mathbb C^n\) with the property \[ B_w(\cdot)\zeta\in\mathcal H,\quad w\in\mathbb D,\;\zeta\in\mathbb C^n, \] \[ \langle f(\cdot),\zeta\rangle=\langle f(\cdot),B_w(\cdot)\zeta\rangle,\quad f\in\mathcal H,\;\zeta\in\mathbb C^n. \]
scientific article; zbMATH DE number 5734656