On Fractional Functional Calculus of Positive Operators

Let N\in B(H) be a normal operator acting on a real or complex Hilbert space H. Define N^\dagger:=N_۱^{-۱}\oplus ۰:\mathcal{R}(N)\oplus \mathcal{K}(N)\rightarrow H, where N_۱=N|_{\mathcal{R}(N)}. Let the {\it fractional semigroup} \mathfrak{F}r(W) denote the collection of all words of the form f_۱^\diamond f_۲^\diamond \cdots f_k^\diamond~ in which ~f_j \in L^\infty (W)~ and ~f^\diamond~ is either ~f~ or ~f^\dagger, where f^\dagger=\chi_{ \{ f\neq ۰ \}}/(f+\chi_{\{f=۰\}}) and L^\infty(W) is a certain normed functional algebra of functions defined on \sigma_\mathbb{F}(W), besides that, W=W^* \in B(H) and \mathbb{F}=\mathbb{R} or \mathbb{C} indicates the underlying scalar field. The {\it fractional calculus} (f_۱^\diamond f_۲^\diamond \cdots f_k^\diamond)(W) on \mathfrak{F}r(W) is defined as f_۱^\diamond(W) f_۲^\diamond (W) \cdots f_k^\diamond (W), where f_j^\dagger(W)=(f_j(W))^\dagger. The present paper studies sufficient conditions on f_j to ensure such fractional calculus are unbounded normal operators. The results will be extended beyond continuous functions.

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عملگر فضا هیلبرتی ، عملگر نرمال بی کران ، حساب تابعی کسری