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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/h/h110/h110120/h1101201.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101202.png" /> is said to be positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101203.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101204.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101205.png" />. In 1951, E. Heinz [[#References|[a3]]] proved a series of very useful norm inequalities; one of the most essential inequalities in operator theory being:
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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 $T\geq0$) if $(Tx,x)\geq0$ for all $x\in H$. In 1951, E. Heinz [[#References|[a3]]] proved a series of very useful norm inequalities; one of the most essential inequalities in operator theory being:
  
<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/h110120/h1101206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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$$\|S_1Q+QS_2\|\geq\|S_1^\alpha QS_2^{1-\alpha}+S_1^{1-\alpha}QS_2^\alpha\|,\tag{a1}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101208.png" /> are positive operators and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h1101209.png" />.
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where $S_1$ and $S_2$ are positive operators and $1\geq\alpha\geq0$.
  
It is shown in [[#References|[a1]]] and [[#References|[a2]]] that the Heinz inequality (a1) is equivalent to each of the inequalities (a2), (a3) and (a4). Other norm inequalities equivalent to (a1) have also been obtained in [[#References|[a1]]] and [[#References|[a2]]]. For any operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h11012010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h11012011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h11012012.png" />,
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It is shown in [[#References|[a1]]] and [[#References|[a2]]] that the Heinz inequality \ref{a1} is equivalent to each of the inequalities \ref{a2}, \ref{a3} and \ref{a4}. Other norm inequalities equivalent to \ref{a1} have also been obtained in [[#References|[a1]]] and [[#References|[a2]]]. For any operators $P$, $Q$ and $R$,
  
<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/h110120/h11012013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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$$\|P^*PQ+QRR^*\|\geq2\|PQR\|.\tag{a2}$$
  
For a self-adjoint and invertible operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h11012014.png" />,
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For a self-adjoint and invertible operator $S$,
  
<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/h110120/h11012015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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$$\|STS^{-1}+S^{-1}TS\|\geq2\|T\|.\tag{a3}$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h11012016.png" /> and self-adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110120/h11012017.png" />,
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For $A\geq0$ and self-adjoint $Q$,
  
<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/h110120/h11012018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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$$\|\operatorname{Re}A^2Q\|\geq\|AQA\|.\tag{a4}$$
  
The inequality (a2) has been obtained in [[#References|[a4]]] to give an alternative ingenious proof of (a1). The original proof of the Heinz inequality (a1), based on deep calculations in complex analysis, is shown in [[#References|[a3]]]; a simplified and elementary proof of (a1) is given in [[#References|[a2]]].
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The inequality \ref{a2} has been obtained in [[#References|[a4]]] to give an alternative ingenious proof of \ref{a1}. The original proof of the Heinz inequality \ref{a1}, based on deep calculations in complex analysis, is shown in [[#References|[a3]]]; a simplified and elementary proof of \ref{a1} is given in [[#References|[a2]]].
  
 
See also [[Heinz–Kato inequality|Heinz–Kato inequality]]; [[Heinz–Kato–Furuta inequality|Heinz–Kato–Furuta inequality]].
 
See also [[Heinz–Kato inequality|Heinz–Kato inequality]]; [[Heinz–Kato–Furuta inequality|Heinz–Kato–Furuta inequality]].

Revision as of 11:16, 9 November 2014

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 $T\geq0$) if $(Tx,x)\geq0$ for all $x\in H$. In 1951, E. Heinz [a3] proved a series of very useful norm inequalities; one of the most essential inequalities in operator theory being:

$$\|S_1Q+QS_2\|\geq\|S_1^\alpha QS_2^{1-\alpha}+S_1^{1-\alpha}QS_2^\alpha\|,\tag{a1}$$

where $S_1$ and $S_2$ are positive operators and $1\geq\alpha\geq0$.

It is shown in [a1] and [a2] that the Heinz inequality \ref{a1} is equivalent to each of the inequalities \ref{a2}, \ref{a3} and \ref{a4}. Other norm inequalities equivalent to \ref{a1} have also been obtained in [a1] and [a2]. For any operators $P$, $Q$ and $R$,

$$\|P^*PQ+QRR^*\|\geq2\|PQR\|.\tag{a2}$$

For a self-adjoint and invertible operator $S$,

$$\|STS^{-1}+S^{-1}TS\|\geq2\|T\|.\tag{a3}$$

For $A\geq0$ and self-adjoint $Q$,

$$\|\operatorname{Re}A^2Q\|\geq\|AQA\|.\tag{a4}$$

The inequality \ref{a2} has been obtained in [a4] to give an alternative ingenious proof of \ref{a1}. The original proof of the Heinz inequality \ref{a1}, based on deep calculations in complex analysis, is shown in [a3]; a simplified and elementary proof of \ref{a1} is given in [a2].

See also Heinz–Kato inequality; Heinz–Kato–Furuta inequality.

References

[a1] J.I. Fujii, M. Fujii, T. Furuta, R. Nakamoto, "Norm inequalities related to McIntosh type inequality" Nihonkai Math. J. , 3 (1992) pp. 67–72
[a2] J.I. Fujii, M. Fujii, T. Furuta, R. Nakamoto, "Norm inequalities equivalent to Heinz inequality" Proc. Amer. Math. Soc. , 118 (1993) pp. 827–830
[a3] E. Heinz, "Beiträge zur Störungstheorie der Spektralzerlegung" Math. Ann. , 123 (1951) pp. 415–438
[a4] A. McIntosh, "Heinz inequalities and perturbation of spectral families" Macquarie Math. Reports (1979) pp. unpublished
How to Cite This Entry:
Heinz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Heinz_inequality&oldid=16400
This article was adapted from an original article by M. Fujii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article