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Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula. (English) Zbl 1070.47009
The authors define and study a smooth functional calculus for a non-commuting tuple of unbounded operators on a Banach space. For any \(f \in C_0^{\infty}(\mathbb{R}^m)\), one can find a function \({\tilde f} \in C_0^{\infty}(\mathbb{C}^m)\) such that \({\tilde f}=f\) on \(\mathbb{R}^m\) and \(\overline{\partial} {\tilde f}=O(| \text{ {Im}} z|^{\infty})\) \(({\overline \partial}=\sum_{j=1}^m \partial\backslash \partial {\overline z}_j d{\overline z}_j )\); one can even assume that \(\text{{supp}} {\tilde f}\) is contained in any small neighborhood of \(\text{{supp}} f\) in \(\mathbb{C}^m\). Let \(P_1, \dots, P_m : {\mathcal B} \to {\mathcal B}\) be densely defined closed operators on the complex Banach space \({\mathcal B}\). Assume that each \(P_j, j=1, \dots, m\), has real spectrum and the resolvents have temperate growth locally near \(\mathbb{R}\): for every compact subset \(K \subset C\) there are \(\mathbb{C}_{K,j}\) and \(N_{K,j} \geq 0\) such that \(\| (z-P_j)^{-1} \| \leq C_{K,j} | \text{ {Im}} z|^{-N_{K,j}}\), \(z \in K\backslash \mathbb{R}\). By definition, \(f(P_1, \dots, P_m)=(-1/\pi)^m \int \dots \int (\partial_{{\overline z}_1} \dots \partial_{{\overline z}_m} {\tilde f}) (z_1-P_1)^{-1} \dots (z_m-P_m)^{-1} L(dz_1) \dots L(dz_m)\), where \(L(dz_j)\) is the Lebesgue measure on \(\mathbb{C}\). This construction is extended to tuples of more general operators allowing smooth functional calculi. The relation to the case of commuting operators is also discussed.

47A60 Functional calculus for linear operators
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