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''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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.G. Krein (ed.) , ''Functional analysis'' , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,   "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.T. Seeley,   "Complex powers of elliptic operators" , ''Proc. Symp. Pure Math.'' , '''10''' , Amer. Math. Soc.  (1967)  pp. 288–307</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hille,   R.S. Phillips,   "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR></table>