Sandberg, Sebastian On non-holomorphic functional calculus for commuting operators. (English) Zbl 1066.32008 Math. Scand. 93, No. 1, 109-135 (2003). 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. Reviewer: Alan J. Pryde (Clayton) Cited in 3 Documents 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 Keywords:functional calculus; resolvent identity Citations:Zbl 0331.47012 PDFBibTeX XMLCite \textit{S. Sandberg}, Math. Scand. 93, No. 1, 109--135 (2003; Zbl 1066.32008) Full Text: DOI