Title: Decomposition of the tensor product of Hilbert modules via the jet construction
Speaker: Soumitra Ghara (IISc Mathematics)
Date: 29 June 2018
Time: 11 am
Venue: LH-1, Mathematics Department

Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ 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 $M_m(\mathbb{C})$ 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 submodules 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 an 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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Last updated: 24 Apr 2019