Multi-parameter transformation groups on white noise functionals
Let \(E\equiv S(R)\subset H\subset E^*\) be a Gelfand triple, \(\mu\) the Gaussian measure on \(E^*\), \((L^2)\equiv L^2(E^*,\mu; \mathbb{C})\), \((E)_\beta\subset (L^2)\subset (E^*)_\beta\) \((\beta\in [0,1))\) a Gelfand triple for \((L^2)\), \((E)_\beta=(E)\) for \(\beta= 0\), and \(E_{\mathbb{C}}\) the complexification of \(E\). \(\phi_\varepsilon(x)= \exp(\langle x,\xi\rangle- \langle\xi,\xi\rangle/2)\in (E)_\beta\) for \(x\in E^*\), \(\xi\in E_{\mathbb{C}}\). The authors define 5-parameter transformations \(H_\alpha(\zeta) \phi_\varepsilon= \alpha_1\exp\{\alpha_2\langle\zeta,\xi\rangle+ \alpha_3\langle\xi, \xi\rangle\}\phi_{\alpha_4\xi+ \alpha_5\zeta}\), for \(\xi,\zeta\in E_{\mathbb{C}}\). If \(\langle\zeta, \zeta\rangle\neq 0\), then \(H= \{H_\alpha(\zeta);\alpha_2, \alpha_3,\alpha_5\in \mathbb{C}, \alpha_1,\alpha_4\in \mathbb{C}\setminus\{0\}\}\) form a subgroup of \(\text{GL}((E))\). They define \(\alpha_i= \alpha_i(\theta; a)\), \(i= 1,\dots, 5\), \(\theta\in\mathbb{R}\), \(a= (a_1, a_2, a_3, a_4, a_5)\in \mathbb{C}^5\) such that \(H_{a;\theta}= H_{\alpha(\theta;a)}(\zeta)\) give a differentiable one parameter subgroup of \(\text{GL}((E))\) with infinitesimal generator \(a_1I+ a_2D_\zeta+ a_3\Delta_G+ a_4 N+ a_5 D_\zeta^*\), \(\Delta_G\) the cross Laplacian operator. For each \(0\neq \zeta\in E_{\mathbb{C}}\), \(h= \langle I,D_\zeta, D_\zeta^*, N,\Delta_G\rangle\) is a 5-dimensional non-nilpotent solvable complex Lie algebra. They define \(K_{(\alpha)}\phi_\xi= \exp\{\sum_{i=1\sim n}\alpha_i\langle\kappa_i, \xi^{\otimes m_i}\rangle\}\phi_{\alpha\xi}\) with \((\alpha)= \alpha,\alpha_1,\dots, \alpha_n\), and study the \((n+1)\)-dimensional transformation subgroup of \(\text{GL}(E)_\beta)\).
non-nilpotent solvable complex Lie algebra
scientific article; zbMATH DE number 1563089