On Fractional Functional Calculus of Positive Operators

Let $Nin B(H)$ be a normal operator acting on a real or complex Hilbert space $H$. Define $N^dagger:=N_1^{-1}oplus 0:mathcal{R}(N)oplus mathcal{K}(N)rightarrow H$, where $N_1=N|_{mathcal{R}(N)}$. Let the {it fractional semigroup} $mathfrak{F}r(W)$ denote the collection of all words of the form $f_1^diamond f_2^diamond cdots f_k^diamond~$ in which $~f_j in L^infty (W)~$ and $~f^diamond~$ is either $~f~$ or $~f^dagger$, where $f^dagger=chi_{ { fneq 0 }}/(f+chi_{{f=0}})$ 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} $(f_1^diamond f_2^diamond cdots f_k^diamond)(W)$ on $mathfrak{F}r(W)$ is defined as $f_1^diamond(W) f_2^diamond (W) cdots f_k^diamond (W)$, 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.