The holomorphic functional calculus as an operational calculus
L. Waelbroeck (1982)
Banach Center Publications
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L. Waelbroeck (1982)
Banach Center Publications
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Mats Andersson, Bo Berndtsson (2002)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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We consider -tuples of commuting operators on a Banach space with real spectra. The holomorphic functional calculus for is extended to algebras of ultra-differentiable functions on , depending on the growth of , , when . In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.
Graham R. Allan (2007)
Banach Center Publications
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This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.
L. Waelbroeck (1983)
Studia Mathematica
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Edoardo Vesentini (1979)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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S. Kowalski, Z. Słodkowski (1980)
Studia Mathematica
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Robin Harte (1996)
Studia Mathematica
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An exactness lemma offers a simplified account of the spectral properties of the "holomorphic" analogue of normal solvability.
Mats Andersson, Sebastian Sandberg (2000)
Studia Mathematica
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We give a new constructive proof of the composition rule for Taylor's functional calculus for commuting operators on a Banach space.
Linus Carlsson, Urban Cegrell, Anders Fällström (2007)
Annales Polonici Mathematici
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We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂̅-problems with certain estimates can be solved. We show that the projection of the spectrum onto ℂⁿ equals Ω̅ and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.