On Fractional Functional Calculus of Positive Operators
Publish place: Wavelets and Linear Algebra، Vol: 8، Issue: 2
Publish Year: 1401
Type: Journal paper
Language: English
View: 92
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JR_WALA-8-2_001
Index date: 5 February 2024
On Fractional Functional Calculus of Positive Operators 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.
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On Fractional Functional Calculus of Positive Operators authors
Moslem Karimzadeh
Department of Mathematics , Kerman Branch , Islamic Azad University , Kerman , Iran
shahrzad azadi
Department of Mathematics , Zahedshahr Branch , Islamic Azad University , Zahedshahr , Iran
Mehdi Radjabalipour
Department of Mathematics , Sh . B . University of Kerman , Kerman , Iran
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