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

References

[a1]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)