Group convolution operators on standard CR-manifolds. I: Structural properties
This paper deals with the convolution operators on \(H_F = \mathbb{R}^m \times \mathbb{C}^n\), endowed with the group operation \((t_1, z_1) \cdot (t_2, z_2) = (t_1 + t_2 + 2 \text{Im }F(z_1, z_2), z_1 + z_2)\), where \((t_k, z_k) \in \mathbb{R}^m \times \mathbb{C}^n\), \(k = 1,2\) and \(F : \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C}^m\) is a Hermitian form. The authors consider the right left group and two-sided convolution operators in the Hilbert space \(L_2(H_F)\) \[ B = (2\pi)^{-N/2} \int_{H_F} b(g) \pi_r (g)dg,\quad C = (2\pi)^{-N/2} \int_{H_F} C(g) \pi_\ell (g) dg, \] \[ K = (2\pi)^{-N} \int_{H_F \times H_F} k(g_1, g_2) \pi_r (g_1) \pi_\ell (g_2) dg_1 dg_2, \] where \(b,c \in L_1(H_F)\), \(k \in L_1(H_F \times H_F)\) and \((\pi_r (g) \varphi) (h) = \varphi (hg)\), \((\pi_\ell (g) \varphi) (h) = \varphi (g^{-1} h)\). The main idea of the investigation is as follows. The operators \(B\), \(C\), \(K\) commute with the shift \(\pi_\ell (t,0)\) on elements \(g = (t,0)\) of the center of \(H_F\) and therefore they also commute with all elements of the von Neumann algebra \(\mathfrak A\) generated by these shift operators. Thus these convolution operators belong to the commutant \({\mathfrak A}' = F^{-1} L_\infty (\mathbb{R}^n)F \otimes {\mathfrak B} (L_2 (\mathbb{C}^n))\) of the algebra \(\mathfrak A\). \(({\mathfrak B} (L_2 (\mathbb{C}^n))\) is an algebra of all bounded linear operators acting on \(L_2(\mathbb{C}^n)\).) Thus all of them have the form \((F^{-1} \otimes I) (I \otimes K(\lambda)) (F \otimes I)\) for some \(K \in L_\infty(\mathbb{R}^n, {\mathfrak B}(L_2(\mathbb{C}^n)))\). The authors obtain formulas for the Weyl symbols of the operators \(K(\lambda)\) and \(K\). Section 3 contains the reduction of \(K(\lambda)\) to some simple canonical form. Section 5 gives a simple isomorphic description of the Hardy space and the Szegö projector associated with the type II Siegel domains.
scientific article; zbMATH DE number 604795
scientific article