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Bounded Perturbations of the Resolvent Operators Associated to the \( \mathcal {F}\)-Spectrum

  • Fabrizio ColomboEmail author
  • Irene Sabadini
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Recently, we have introduced the F-functional calculus and the SC-functional calculus. Our theory can be developed for operators of the form T = T 0 + e 1 T 1 +...+ e n T n where (T 0, T 1,...,T n) is an (n + 1)-tuple of linear commuting operators. The SC-functional calculus, which is defined for bounded but also for unbounded operators, associates to a suitable slice monogenic function f with values in the Clifford algebra ℝn the operator f(T). The F-functional calculus has been defined, for bounded operators T, by an integral transform. Such an integral transform comes from the Fueter’s mapping theorem and it associates to a suitable slice monogenic function f the operator \(\breve{f} (T)\), where \(\breve{f}(x)=\Delta^{\frac{n-1}{2}}f(x)\) and Δ is the Laplace operator. Both functional calculi are based on the notion of F-spectrum that plays the role that the classical spectrum plays for the Riesz-Dunford functional calculus. The aim of this paper is to study the bounded perturbations of the SC-resolvent operator and of the F-resolvent operator. Moreover we will show some examples of equations that lead to the F-spectrum.

Keywords

Functional calculus for n-tuples of commuting operators F-spectrum perturbation of the SC-resolvent operator perturbation of the F-resolvent operator examples of equations that lead to the F-spectrum. 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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