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''of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412701.png" /> on a complex Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412702.png" />''
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$#C+1 = 78 : ~/encyclopedia/old_files/data/F041/F.0401270 Fractional power
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412703.png" /> of this operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412704.png" />. If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412705.png" /> is bounded and its spectrum does not contain zero and does not surround it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412706.png" /> is defined by a [[Cauchy integral|Cauchy integral]] along a contour around the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412707.png" /> not containing zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412708.png" /> is unbounded, the contour has to be taken infinite, and problems on the convergence of the integral arise. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f0412709.png" /> has a domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127010.png" /> which is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127011.png" />, and has for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127012.png" /> a resolvent
+
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 +
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127013.png" /></td> </tr></table>
+
''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
 
satisfying the estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\| R ( - s , A ) \|  \leq  M ( 1 + s ) ^ {-} 1 ,\  s > 0 ,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127015.png" /></td> </tr></table>
+
$$
 +
A ^ {- \alpha }  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma  \lambda ^ {-
 +
\alpha } R ( \lambda , A )  d \lambda ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127016.png" /> consists of the sides of an angle depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127017.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127018.png" /> are bounded and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127019.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127020.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127021.png" />. Positive powers are defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127022.png" />; they are unbounded. For any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127024.png" /> the following fundamental property of powers holds:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127025.png" /></td> </tr></table>
+
$$
 +
A  ^  \alpha  A  ^  \beta  x  = A  ^  \beta  A  ^  \alpha  x  = A ^ {
 +
\alpha + \beta } x
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127029.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127031.png" />,
+
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  ) $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127032.png" /></td> </tr></table>
+
$$
 +
\| A  ^  \beta  x \|  \leq  C ( \alpha , \beta , \gamma ) \| A  ^  \alpha  x \| ^ {( \gamma - \beta ) / ( \gamma - \alpha ) } \| A  ^  \gamma  x \| ^ {( \beta - \alpha ) / ( \gamma - \alpha ) }
 +
$$
  
(inequality of moments). The power semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127033.png" /> permits extension to the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127034.png" /> which is analytic in the right half-plane.
+
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127035.png" /> has no bounded inverse and when the estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127037.png" />, holds. If condition (1) is met and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127038.png" />, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127039.png" /></td> </tr></table>
+
$$
 +
A ^ {- \alpha }  =
 +
\frac{\sin  \alpha \pi } \pi
 +
\int\limits _ { 0 } ^  \infty  s ^ {- \alpha } R ( - s , A )  d s .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127040.png" /> is the infinitesimal operator of a contraction semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127041.png" />, then
+
If $  B $
 +
is the infinitesimal operator of a contraction semi-group $  U ( t) $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127042.png" /></td> </tr></table>
+
$$
 +
( - B ) ^ {- \alpha }  =
 +
\frac{1}{\Gamma ( \alpha ) }
 +
\int\limits _ { 0 } ^  \infty  t ^ {\alpha - 1 } U ( t)  d t .
 +
$$
  
It does not follow from condition (1) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127043.png" /> is the infinitesimal operator of a strongly-continuous semi-group, but the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127044.png" /> is the infinitesimal operator of an analytic semi-group if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127045.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127046.png" /> is dominated by an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127047.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127048.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127050.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127051.png" /> is dominated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127052.png" /> and if the resolvents of both operators have the property (1), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127053.png" /> is dominated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127054.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127055.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127056.png" /> 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]]):
+
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]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127057.png" /></td> </tr></table>
+
$$
 +
A  ^  \alpha  = \int\limits _ { 0 } ^  \infty  \lambda  ^  \alpha  d E _  \lambda  .
 +
$$
  
In the inequality of moments, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127058.png" /> for such an operator. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127060.png" /> be two positive self-adjoint operators, acting on Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127062.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127063.png" /> is a bounded linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127064.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127065.png" /> with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127066.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127069.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127070.png" /> and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127071.png" /></td> </tr></table>
+
$$
 +
\| B  ^  \alpha  T x \|  \leq  M ^ {1 - \alpha } M _ {1}  ^  \alpha
 +
\| A  ^  \alpha  x \| ,\  0 \leq  \alpha \leq  1
 +
$$
  
(Heinz's inequality). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127073.png" />, the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127074.png" /> is dominated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127075.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127076.png" /> is dominated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127077.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041270/f04127078.png" />. 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.
+
(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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:39, 5 June 2020


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

Comments

References

[a1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
How to Cite This Entry:
Fractional power. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional_power&oldid=18216
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article