| Popis: |
Given a pair of positive real numbers α, β and a sesqui-analytic function K on a bounded domain \(\Omega \subset \mathbb C^m\), in this paper, we investigate the properties of the sesqui-analytic function $$\displaystyle \mathbb K^{(\alpha , \beta )}:= K^{\alpha +\beta }\big (\partial _i\bar {\partial }_j\log K\big )_{i,j=1}^ m $$ taking values in m × m matrices. One of the key findings is that \(\mathbb K^{(\alpha , \beta )}\) is non-negative definite whenever Kα and Kβ are non-negative definite. In this case, a realization of the Hilbert module determined by the kernel \(\mathbb K^{(\alpha ,\beta )}\) is obtained. Let \(\mathcal M_i\), i = 1, 2, be two Hilbert modules over the polynomial ring \(\mathbb C[z_1, \ldots , z_m]\). Then \(\mathbb C[z_1, \ldots , z_{2m}]\) acts naturally on the tensor product \(\mathcal M_1\otimes \mathcal M_2\). The restriction of this action to the polynomial ring \(\mathbb C[z_1, \ldots , z_m]\) obtained using the restriction map p↦p| Δ leads to a natural decomposition of the tensor product \(\mathcal M_1\otimes \mathcal M_2\), which is investigated. Two of the initial pieces in this decomposition are identified. |
