TY - JOUR ID - 251146 TI - On Fractional Functional Calculus of Positive Operators JO - Wavelet and Linear Algebra JA - WALA LA - en SN - 2383-1936 AU - Karimzadeh, Moslem AU - azadi, shahrzad AU - Radjabalipour, Mehdi AD - Department of Mathematics‎, ‎Kerman Branch‎, ‎Islamic Azad University‎, ‎Kerman‎, ‎Iran AD - Department of Mathematics‎, ‎Zahedshahr Branch‎, ‎Islamic Azad University‎, ‎Zahedshahr‎, ‎Iran AD - Department of Mathematics‎, ‎Sh‎. ‎B‎. ‎University of Kerman‎, ‎Kerman‎, ‎Iran Y1 - 2022 PY - 2022 VL - 8 IS - 2 SP - 1 EP - 9 KW - Hilbert space operator KW - Unbounded normal operator KW - Fractional functional calculus DO - 10.22072/wala.2021.525358.1320 N2 - 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. UR - https://wala.vru.ac.ir/article_251146.html L1 - https://wala.vru.ac.ir/article_251146_75c9792fa131e0bfc30709a00b07bf7a.pdf ER -