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In the sequel, a capital letter denotes a bounded [[Linear operator|linear operator]] on a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101901.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101902.png" /> is said to be positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101903.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101904.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101905.png" />.
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A real-valued continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101906.png" /> is called an operator-monotone function if
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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/l/l110/l110190/l1101907.png" /></td> </tr></table>
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In the sequel, a capital letter denotes a bounded [[Linear operator|linear operator]] on a [[Hilbert space|Hilbert space]]  $  H $.
 +
An operator  $  T $
 +
is said to be positive (denoted by  $  T \geq  0 $)
 +
if  $  ( {Tx } , x ) \geq  0 $
 +
for all  $  x \in H $.
  
K. Löwner [[#References|[a3]]] has completely characterized all operator-monotone functions as the class of Pick functions. These functions play an essential and very important role in the theory of analytic functions. He has also proved that a real-valued continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101908.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l1101909.png" /> is operator monotone if and only if it has a representation
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A real-valued continuous function $  f $
 +
is called an operator-monotone function if
  
<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/l/l110/l110190/l11019010.png" /></td> </tr></table>
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$$
 +
A \geq  B \geq  0 \Rightarrow  f ( A ) \geq  f ( B ) .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019012.png" /> are two arbitrary real constants. This representation shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019013.png" /> is operator monotone if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019014.png" /> and it is not operator monotone if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019015.png" />. This fact can be expressed as follows (the Löwner–Heinz inequality, 1934): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019016.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110190/l11019018.png" />.
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K. Löwner [[#References|[a3]]] has completely characterized all operator-monotone functions as the class of Pick functions. These functions play an essential and very important role in the theory of analytic functions. He has also proved that a real-valued continuous function  $  f $
 +
on  $  ( 0, \infty ) $
 +
is operator monotone if and only if it has a representation
 +
 
 +
$$
 +
f ( t ) = at + b + \int\limits _ { 0 } ^  \infty  {
 +
\frac{t}{t + s }
 +
}  d \mu ( s ) ,
 +
$$
 +
 
 +
where  $  a \geq  0 $
 +
and  $  b $
 +
are two arbitrary real constants. This representation shows that $  f ( t ) = t  ^  \alpha  $
 +
is operator monotone if $  \alpha \in [ 0,1 ] $
 +
and it is not operator monotone if $  \alpha > 1 $.  
 +
This fact can be expressed as follows (the Löwner–Heinz inequality, 1934): $  A \geq  B \geq  0 $
 +
implies $  A  ^  \alpha  \geq  B  ^  \alpha  $
 +
for all $  \alpha \in [ 0,1 ] $.
  
 
The Löwner–Heinz inequality is not only the most famous and profound one in operator theory, but also a useful and fundamental tool for treating operator inequalities. The original proof in [[#References|[a3]]] used the above integral representation for operator-monotone functions; an alternative proof was been discovered in 1951 (cf. [[#References|[a1]]]), while a short proof can be found in [[#References|[a2]]]. Various different proofs of the Löwner–Heinz inequality have been given by many authors. There are proofs based on an analytical method in the theory of differential equations as well as simple algebraic proofs. A breathtakingly elegant proof can be found in [[#References|[a4]]]. (This is a motivation of the [[Furuta inequality|Furuta inequality]].) It is known that the Löwner–Heinz inequality, the [[Heinz–Kato inequality|Heinz–Kato inequality]] and the Cordes inequality are mutually equivalent, although the first is an operator inequality while the latter two are norm inequalities. Moreover, they are also equivalent to the Jensen inequality:
 
The Löwner–Heinz inequality is not only the most famous and profound one in operator theory, but also a useful and fundamental tool for treating operator inequalities. The original proof in [[#References|[a3]]] used the above integral representation for operator-monotone functions; an alternative proof was been discovered in 1951 (cf. [[#References|[a1]]]), while a short proof can be found in [[#References|[a2]]]. Various different proofs of the Löwner–Heinz inequality have been given by many authors. There are proofs based on an analytical method in the theory of differential equations as well as simple algebraic proofs. A breathtakingly elegant proof can be found in [[#References|[a4]]]. (This is a motivation of the [[Furuta inequality|Furuta inequality]].) It is known that the Löwner–Heinz inequality, the [[Heinz–Kato inequality|Heinz–Kato inequality]] and the Cordes inequality are mutually equivalent, although the first is an operator inequality while the latter two are norm inequalities. Moreover, they are also equivalent to the Jensen inequality:
  
<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/l/l110/l110190/l11019019.png" /></td> </tr></table>
+
$$
 +
( X  ^ {*} A X )  ^  \alpha  \geq  X  ^ {*} A X
 +
$$
  
<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/l/l110/l110190/l11019020.png" /></td> </tr></table>
+
$$
 +
\textrm{ for  }  A \geq  0,  \textrm{ contractions  }  X,  \alpha \in [ 0,1 ] .
 +
$$
  
 
See also [[Furuta inequality|Furuta inequality]].
 
See also [[Furuta inequality|Furuta inequality]].

Latest revision as of 04:11, 6 June 2020


In the sequel, a capital letter denotes a bounded linear operator on a Hilbert space $ H $. An operator $ T $ is said to be positive (denoted by $ T \geq 0 $) if $ ( {Tx } , x ) \geq 0 $ for all $ x \in H $.

A real-valued continuous function $ f $ is called an operator-monotone function if

$$ A \geq B \geq 0 \Rightarrow f ( A ) \geq f ( B ) . $$

K. Löwner [a3] has completely characterized all operator-monotone functions as the class of Pick functions. These functions play an essential and very important role in the theory of analytic functions. He has also proved that a real-valued continuous function $ f $ on $ ( 0, \infty ) $ is operator monotone if and only if it has a representation

$$ f ( t ) = at + b + \int\limits _ { 0 } ^ \infty { \frac{t}{t + s } } d \mu ( s ) , $$

where $ a \geq 0 $ and $ b $ are two arbitrary real constants. This representation shows that $ f ( t ) = t ^ \alpha $ is operator monotone if $ \alpha \in [ 0,1 ] $ and it is not operator monotone if $ \alpha > 1 $. This fact can be expressed as follows (the Löwner–Heinz inequality, 1934): $ A \geq B \geq 0 $ implies $ A ^ \alpha \geq B ^ \alpha $ for all $ \alpha \in [ 0,1 ] $.

The Löwner–Heinz inequality is not only the most famous and profound one in operator theory, but also a useful and fundamental tool for treating operator inequalities. The original proof in [a3] used the above integral representation for operator-monotone functions; an alternative proof was been discovered in 1951 (cf. [a1]), while a short proof can be found in [a2]. Various different proofs of the Löwner–Heinz inequality have been given by many authors. There are proofs based on an analytical method in the theory of differential equations as well as simple algebraic proofs. A breathtakingly elegant proof can be found in [a4]. (This is a motivation of the Furuta inequality.) It is known that the Löwner–Heinz inequality, the Heinz–Kato inequality and the Cordes inequality are mutually equivalent, although the first is an operator inequality while the latter two are norm inequalities. Moreover, they are also equivalent to the Jensen inequality:

$$ ( X ^ {*} A X ) ^ \alpha \geq X ^ {*} A X $$

$$ \textrm{ for } A \geq 0, \textrm{ contractions } X, \alpha \in [ 0,1 ] . $$

See also Furuta inequality.

References

[a1] E. Heinz, "Beiträge zur Störungstheorie der Spektralzerlegung" Math. Ann. , 123 (1951) pp. 415–438
[a2] T. Kato, "Notes on some inequalities for linear operators" Math. Ann. , 125 (1952) pp. 208–212
[a3] K. Löwner, "Über monotone Matrixfunktionen" Math. Z. , 38 (1934) pp. 177–216
[a4] G.K. Pedersen, "Some operator monotone functions" Proc. Amer. Math. Soc. , 36 (1972) pp. 309–310
How to Cite This Entry:
Löwner-Heinz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%B6wner-Heinz_inequality&oldid=23406
This article was adapted from an original article by M. Fujii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article