Hypercomplex Analysis and Applications pp 13-28 | Cite as

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

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## 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 ℝ*n* 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 S_{C}-resolvent operator perturbation of the F-resolvent operator examples of equations that lead to the F-spectrum.

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