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On non-holomorphic functional calculus for commuting operators. (English) Zbl 1066.32008

Via the Cauchy-Green formula, the Riesz functional calculus for an operator \(a\) on a Banach space \(X\) was extended by E. M. Dynkin [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 30, 33–39 (1972; Zbl 0331.47012)] to an algebra \(S_{a}\) of non-holomorphic functions. Indeed, \(S_{a}\) is the space of functions \(f\in C_{c}^{1}(\mathbb C)\) with \(\| \overline{\partial }f(z)\wedge (z-a)^{-1}\,dz\| _{\infty }<\infty \) and \(f(a)=-\frac{1}{2\pi i}\int \overline{\partial }f(z)\wedge (z-a)^{-1}\,dz\). The map \(f\mapsto f(a):S_{a}\rightarrow L(X)\) is a continuous algebra homomorphism and this functional calculus satisfies the spectral mapping theorem \(\sigma (f(a))=f(\sigma (a))\). For commuting tuples of operators, J. L. Taylor used the Koszul complex to define a joint spectrum and the Cauchy-Weil integral to define a holomorphic functional calculus satisfying the spectral mapping theorem.
Following Dynkin, the paper extends the Taylor functional calculus to an algebra of non-holomorphic functions. Again the spectral mapping theorem holds. A key step is a generalized resolvent identity for several commuting operators.

MSC:

32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
47A10 Spectrum, resolvent
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A60 Functional calculus for linear operators
32A38 Algebras of holomorphic functions of several complex variables

Citations:

Zbl 0331.47012
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