On multidimensional Bochner-Phillips functional calculus (Q2888242)

Let \({\mathcal T}_n\) be the class of functions \(\psi \in C^{\infty}((-\infty,0)^n)\) with \(\partial^{\alpha}\psi \geq 0\) for all multi-indices \(\alpha\). Each function \(\psi \in {\mathcal T}_n\) allows the integral representation \(\psi(s)=c_0+c_1\cdot s+\int_{\mathbb R^n_{+} \backslash \{ 0 \}} (e^{s\cdot u}-1) \, d\mu (u)\) with \(c_0=\psi(-0)\), \(c_1 \in \mathbb R^n_{+}\), and a positive measure \(\mu\) on \(\mathbb R^n_{+}\backslash \{ 0 \}\). Suppose that we are given commuting bounded \(C_0\)-semigroups \(T_1, \dots, T_n\) in a complex Banach space \(X\) with generators \(A=(A_1,\dots,A_n)\). The author studies the multidimensional operator calculus defined by \(\psi(A)x=c_0 x+c_1\cdot Ax+ \int_{\mathbb R^n_{+}\backslash \{ 0 \}} (T(u)-I)x \, d\mu(u)\), where \(T(u)=(T_1(u_1),\dots,T_n(u_n))\), \(c_1\cdot Ax=\sum_{j=1}^n c_1^j A_j x\), and \(D(A)=\cap_{j=1}^n D(A_j)\).