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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.

MSC:
47A60 Functional calculus for linear operators
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