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Non-holomorphic functional calculus for commuting operators with real spectrum. (English) Zbl 1099.47505

Let \(X\) be a Banach space and let \(a=(a_1, a_2, \dots, a_n)\) be an \(n\)-tuple of bounded linear commuting operators on \(X\). Then \(f(a)\) has a definite meaning for every polynomial \(f\), and this functional calculus can be extended to a continuous algebra homomorphism from the space of entire functions on \(\mathbb C^n\) into \(L(X)\). To obtain a functional calculus on a larger class than the space of entire functions, one needs to consider the joint spectrum of the operators. For the case when this spectrum is real, the authors show that the functional calculus can be extended to algebras of ultra-differentiable functions, depending on the growth of \(\left\| \exp(ia \cdot t) \right\|\) as \(| t | \to \infty\). In the non-quasi-analytic case, they use the usual Fourier transform, whereas in the quasi-analytic case they introduce a suitable variant of the FBI transform.

MSC:

47A60 Functional calculus for linear operators
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A65 Banach algebra techniques applied to functions of several complex variables
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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References:

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