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

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.

Mathematics Subject Classification (2010). Primary 30G35; Secondary 47A10.