Let K be a bounded domain and K:\Omega \times \Omega \to C be a sesqui-analytic function. We show that if \alpha,\beta>0 be such that the functions K^{\alpha} and K^{\beta}, defined on \Omega\times\Omega, are non-negative definite kernels, then the M_m(C) valued function

K^{(\alpha,\beta)}: =K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^ m is also a non-negative definite kernel on \Omega\times\Omega. Then we find a realization of the Hilbert space (H,K^{(\alpha,\beta)})determined by the kernel K^{(\alpha, \beta)} in terms of the tensor product (H, K^{\alpha})\otimes (H, K^{\beta}).

For two reproducing kernel Hilbert modules (H,K_1) and (H,K_2), let A_n, n\geq 0, be the submodule of the Hilbert module (H, K_1)\otimes (H, K_2) consisting of functions vanishing to order n on the diagonal set \Delta:={(z,z):z\in \Omega}. Setting S_0=A_0^\perp, S_n=A_{n-1}\ominus A_{n}, n\geq 1, leads to a natural decomposition of (H, K_1)\otimes (H, K_2) into infinite direct sum \oplus{n=0}^{\infty} S_n. A theorem of Aronszajn shows that the module S_0 is isometrically isomorphic to the push-forward of the module (H,K_1K_2) under the map \iota:\Omega\to \Omega\times\Omega, where iota(z)=(z,z), z\in \Omega. We prove that if K_1=K^{\alpha} and K_2=K^{\beta}, then the module S_1 is isometrically isomorphic to the push-forward of the module (H,K^{(\alpha, \beta)}) under the map \iota. We also show that if a scalar valued non-negative kernel K is quasi-invariant, then K^{(1,1)} is also a quasi-invariant kernel.

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