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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
##### Keywords:
functional calculus; spectrum; non-commuting operators
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##### References:
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