Recently several operators are defined with having the keyword "fractional derivative".
My question is that what properties should an operator have to call it "fractional"? Additionally, when we can say an operator is a derivative?
what is the characteristic property for fractional derivatives?
Among the newly defined operators for derivatives with non-integer order which one is "fractional" and which one can be called a "derivative" operator?
Generally, the definition of the fractional power of an operator in an abstract space follows via Dunford's integral. Indeed, the case of fractional derivative of a function is similar.
The derivative is well-known, and you can define the derivative on some space as operator, you need then a norm.
For the fractional Leibniz' rule see the attachments. The first reference is from the book of Igor Podlubny (1999) : Fractional Differential Equations. See also chapter 4 of the paper of Ehsan Azmoodeh : Riemann-Stieltjes integrals with respect to function Brownian motion and applications. You can see there also the fractional integration by parts.
Fractional derivatives are partial derivatives dependent upon direction.
The fraction derivation, in operator theory, is the converse of the fractionnal integration operation. The latter one is defined as an operator on functionnal space mainly, as prototype, $ C([0,1]).$ More precise, it is defined for all $f\in C([0,1])$ and all $0
Above, you define the fractional power of integral. So, it is better if you use $I^\{\alpha}$ instead of that with D. Then it follows ${I^{\alpha}}^{-1}}$=$D^{\alpha}$.
Ok as you like. indeed in the litterature, the notation you suggest is usually used. you are right
The two attached papers, I believe, should aid in understanding the operation and geometric meaning of fractional order derivatives.
Personally, I like to think of fractional derivation or integration in terms of Fourier theory, where no limit exist as to how many multiple splices one mapping may be split.
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