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Multicentric holomorphic calculus. (English) Zbl 1337.47021

Let \(A\) be a complex unital Banach algebra. For \(a\in A\), let \(\widehat{\sigma(a)}\) denote the polynomially convex hull of the spectrum of \(a\). Let \(\Omega\subset \mathbb{C}\) be an open set such that \(\widehat{\sigma(a)}\subset\Omega\). If \(f\) is holomorphic in \(\Omega\) and \(\Gamma\subset\Omega\) is a contour surrounding \(\widehat{\sigma(a)}\), then \[ f(a) = \frac{1}{2\pi i} \int_{\Gamma} f(\lambda) (\lambda-a)^{-1} d\lambda. \]
The author constructs a practical power series calculus based on approximations of the resolvent \((\lambda-a)^{-1}\) by rational functions. This approach was outlined by the author in [Numer. Funct. Anal. Optim. 30, No.  9–10, 1025–1047 (2009; Zbl 1188.47005)].

MSC:

47A60 Functional calculus for linear operators
46H30 Functional calculus in topological algebras
47A10 Spectrum, resolvent
30B99 Series expansions of functions of one complex variable
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators

Citations:

Zbl 1188.47005
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Full Text: DOI

References:

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