Section 4.8 Fractional Powers
The above spectral representations provide a natural way to define fractional operators such as \(\sqrt{-\Delta}\) for example. Fractional operators are feared by some and loved by others. They are popular in the analysis of non-standard random walks such as Levy flights \cite{Levyflights}. The analysis of fractional power operators is tricky, and the spectral representation below gives us a powerful tool for their analysis.Definition 4.8.1.
If a positive operator \(A\) on a Hilbert space has a representation as
\begin{equation*}
Au=\sum_{j=1}^\infty \lambda_j(u,w_j)w_j
\end{equation*}
then we define the fractional powers of \(A\) as
\begin{equation*}
A^\alpha u =\sum_{j=1}^\infty \lambda_j^\alpha (u,w_j)w_j
\end{equation*}
for \(\alpha \in \mathbb{R}, \text{ }\alpha\geq 0\) with domain
\begin{equation}
D(A^\alpha)=\left\{u: \|A^\alpha u\|<\infty\right\}=\left\{u: u=\sum_{j=1}^\infty c_jw_j, \quad \sum_{j=1}^\infty | c_j|^{2}\lambda_j^{2\alpha}<\infty \right\}.\tag{4.8.1}
\end{equation}
\begin{equation*}
(u,v)_{D(A^\alpha)}:=\left(A^\alpha u , A^\alpha v\right)
\end{equation*}
with the norm
\begin{equation*}
\|u\|_{D(A^\alpha)}=\|A^\alpha u\|.
\end{equation*}
Example 4.8.2.
\begin{equation}
D(A^\frac{1}{2})=\left\{u=\sum c_jw_j, \quad \sum_{j=1}^\infty\mid c_j\mid\lambda_j^2<\infty\right\}\quad\mbox{ and norm } \quad
\|u\|_\frac{1}{2}=\sum_{j=1}^\infty\mid c_j\mid\lambda_j^2.\tag{4.8.2}
\end{equation}