On Fractional Functional Calculus of Positive Operators

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Zahedshahr Branch‎, ‎Islamic Azad University‎, ‎Zahedshahr‎, ‎Iran

2 Department of Mathematics‎, ‎Kerman Branch‎, ‎Islamic Azad University‎, ‎Kerman‎, ‎Iran

3 Department of Mathematics‎, ‎Sh‎. ‎B‎. ‎University of Kerman‎, ‎Kerman‎, ‎Iran

10.22072/wala.2021.525358.1320

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.

Keywords


[1] S. Azadi and M. Radjabalipour, Algebraic frames and duality, J. Math. Ext., 15(3), (2021), (9)1-11.
[2] S. Azadi and M. Radjabalipour, On the structure of unbounded linear operators, Iran. J. Sci. Technol., Trans. A, Sci.
     44(6), (2020), 1711-1719.
[3] Y.M. Brezansky, Z.G. Sheftel and G.F. Us, Functional Analysis; Vol. II, Birkh$ddot{a}$user Verlag, Basel, 1996.
[4] J.B. Conway, A Course in Functional Analysis, 2nd Edition, Springer-Verlag, New York, 1997.
[5] N. Dunford and J.T. Schwartz, Linear Operators; Part I: General Theory, Interscience Publishers, New York, 1957.
[6] N. Dunford and J.T. Schwartz, Linear Operators; Part II:Spectral Theory, Selfadjoint Operators in Hilbert Space
      Interscience Publishers, New York, 1963.
[7] P.M. Fitzpatrick, A note on the functional calculus for unbounded selfadjoint operators, J. Fixed Point Theor. Appl., 13
      (2013), 633-640.
[8] W. Groetsch, Stable Approximate Evaluation of Unbounded Operators, Springer, New York, 2006.
[9] M. Karimzadeh and M. Radjabalipour, On properties of real selfadjoint operators, Banach J. Math. Anal., 15(1), (2020), 
     DOI: 10.1007/s43037-020-00101-x.
[10] G.K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989.
[11] M. Radjabalipour, On Fitzpatrick functions of monotone linear operators, J. Math. Anal. Appl., 401, (2013), 950--958.