Fractional power

of a linear operator $A$ on a complex Banach space $E$

A function $f ( A)$ of this operator such that $f ( z) = z ^ \alpha$. If the operator $A$ is bounded and its spectrum does not contain zero and does not surround it, $A ^ \alpha$ is defined by a Cauchy integral along a contour around the spectrum of $A$ not containing zero. If $A$ is unbounded, the contour has to be taken infinite, and problems on the convergence of the integral arise. If $A$ has a domain of definition $D ( A)$ which is dense in $E$, and has for $\lambda < 0$ a resolvent

$$R ( \lambda , A ) = ( A - \lambda I ) ^ {-} 1$$

satisfying the estimate

$$\tag{1 } \| R ( - s , A ) \| \leq M ( 1 + s ) ^ {-} 1 ,\ s > 0 ,$$

then

$$A ^ {- \alpha } = \frac{1}{2 \pi i } \int\limits _ \Gamma \lambda ^ {- \alpha } R ( \lambda , A ) d \lambda ,$$

where $\Gamma$ consists of the sides of an angle depending on $M$. The operators $A ^ {- \alpha }$ are bounded and $A ^ {- \alpha } x \rightarrow x$ for any $x \in E$ as $\alpha \rightarrow 0$. Positive powers are defined as follows: $A ^ \alpha = ( A ^ {- \alpha } ) ^ {-} 1$; they are unbounded. For any real $\alpha$ and $\beta$ the following fundamental property of powers holds:

$$A ^ \alpha A ^ \beta x = A ^ \beta A ^ \alpha x = A ^ { \alpha + \beta } x$$

for $x \in D ( A ^ \gamma )$ and $\gamma = \max \{ \alpha , \beta , \alpha + \beta \}$. If $0 < \alpha < 1$, $( A ^ \alpha ) ^ \beta = A ^ {\alpha \beta }$. For any $\alpha < \beta < \gamma$ and $x \in D ( A ^ \gamma )$,

$$\| A ^ \beta x \| \leq C ( \alpha , \beta , \gamma ) \| A ^ \alpha x \| ^ {( \gamma - \beta ) / ( \gamma - \alpha ) } \| A ^ \gamma x \| ^ {( \beta - \alpha ) / ( \gamma - \alpha ) }$$

(inequality of moments). The power semi-group $A ^ {- \alpha }$ permits extension to the semi-group $A ^ {-} z$ which is analytic in the right half-plane.

The above properties are extended to include the case when $A$ has no bounded inverse and when the estimate $\| R ( - s , A ) \| \leq M s ^ {-} 1$, $s > 0$, holds. If condition (1) is met and if $0 < \alpha < 1$, then

$$A ^ {- \alpha } = \frac{\sin \alpha \pi } \pi \int\limits _ { 0 } ^ \infty s ^ {- \alpha } R ( - s , A ) d s .$$

If $B$ is the infinitesimal operator of a contraction semi-group $U ( t)$, then

$$( - B ) ^ {- \alpha } = \frac{1}{\Gamma ( \alpha ) } \int\limits _ { 0 } ^ \infty t ^ {\alpha - 1 } U ( t) d t .$$

It does not follow from condition (1) that $- A$ is the infinitesimal operator of a strongly-continuous semi-group, but the operator $- A ^ \alpha$ is the infinitesimal operator of an analytic semi-group if $\alpha \leq 1/2$.

An operator $B$ is dominated by an operator $A$ if $D ( B) \supset D ( A)$ and if $\| Bx \| \leq c \| Ax \|$, $x \in D ( A)$. If $B$ is dominated by $A$ and if the resolvents of both operators have the property (1), then $B ^ \alpha$ is dominated by $A ^ \beta$ if $0 \leq \alpha < \beta \leq 1$.

If $A$ is a positive self-adjoint operator on a Hilbert space, its fractional power is defined by the spectral decomposition (cf. Spectral decomposition of a linear operator):

$$A ^ \alpha = \int\limits _ { 0 } ^ \infty \lambda ^ \alpha d E _ \lambda .$$

In the inequality of moments, $c ( \alpha , \beta , \gamma ) = 1$ for such an operator. Let $A$ and $B$ be two positive self-adjoint operators, acting on Hilbert spaces $H$ and $H _ {1}$, respectively. If $T$ is a bounded linear operator from $H$ to $H _ {1}$ with norm $M$ such that $T D ( A) \subset D ( B)$ and $\| B T x \| \leq M _ {1} \| A x \|$, $x \in D ( A)$, then $T D ( A ^ \alpha ) \subset D ( B ^ \alpha )$ and

$$\| B ^ \alpha T x \| \leq M ^ {1 - \alpha } M _ {1} ^ \alpha \| A ^ \alpha x \| ,\ 0 \leq \alpha \leq 1$$

(Heinz's inequality). In particular, if $H = H _ {1}$ and $T = I$, the fact that $B$ is dominated by $A$ implies that $B ^ \alpha$ is dominated by $A ^ \alpha$ if $0 \leq \alpha \leq 1$. Fractional powers of operators are employed in the study of non-linear equations. They have been studied in detail for operators generated by elliptic boundary value problems.

References

 [1] S.G. Krein (ed.) , Functional analysis , Wolters-Noordhoff (1972) (Translated from Russian) [2] S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) [3] R.T. Seeley, "Complex powers of elliptic operators" , Proc. Symp. Pure Math. , 10 , Amer. Math. Soc. (1967) pp. 288–307