Wavelet and Linear Algebra

Abstract Let $N\in B(H)$ 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 $f^\dagger=\chi_{ \{ f\neq 0 \}}/(f+\chi_{\{f=0\}})$ 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.