View source for Heinz-Kato inequality

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/h/h110/h110140/h1101401.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h1101402.png" /> is said to be positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h1101403.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h1101404.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h1101405.png" />.

The Heinz–Kato inequality is an extension of the generalized Cauchy–Schwarz inequality (cf. also [[Cauchy inequality|Cauchy inequality]]). It follows from the fact that

<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/h/h110/h110140/h1101406.png" /></td> </tr></table>

<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/h/h110/h110140/h1101407.png" /></td> </tr></table>

is non-negative, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h1101408.png" /> is the [[Polar decomposition|polar decomposition]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h1101409.png" />.

The Heinz–Kato inequality (1952; cf. [[#References|[a4]]], [[#References|[a3]]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014011.png" /> are positive operators such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014014.png" />, then the following inequality holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014015.png" />:

<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/h/h110/h110140/h11014016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014017.png" />.

It is proved in [[#References|[a1]]] that the Heinz–Kato inequality is equivalent to:

<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/h/h110/h110140/h11014018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

for arbitrary positive operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014020.png" />.

The [[Heinz inequality|Heinz inequality]] yields the Heinz–Kato inequality.

On the other hand, it is shown in [[#References|[a2]]] that the [[Löwner–Heinz inequality|Löwner–Heinz inequality]] is equivalent to the following Cordes inequality (a3), although the first is an operator inequality and the latter is a norm 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/h/h110/h110140/h11014021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110140/h11014023.png" />.

It is well known that the Heinz–Kato inequality (a1) is equivalent to the Löwner–Heinz inequality, so that the Heinz–Kato inequality, the Löwner–Heinz inequality and the Cordes inequality are mutually equivalent.

Additional references can be found in [[Heinz inequality|Heinz inequality]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Fujii,   T. Furuta,   "Löwner–Heinz, Cordes and Heinz–Kato inequalities"  ''Math. Japon.'' , '''38'''  (1993)  pp. 73–78</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Furuta,   "Norm inequalities equivalent to Löwner–Heinz theorem"  ''Rev. Math. Phys.'' , '''1'''  (1989)  pp. 135–137</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Kato,   "Notes on some inequalities for linear operators"  ''Math. Ann.'' , '''125'''  (1952)  pp. 208–212</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Heinz,   "Beiträge zur Störungstheorie der Spektralzerlegung"  ''Math. Ann.'' , '''123'''  (1951)  pp. 415–438</TD></TR></table>