Difference between revisions of "Heinz-Kato inequality"
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| − | In the sequel, a capital letter denotes a bounded [[Linear operator|linear operator]] on a [[Hilbert space|Hilbert space]] | + | <!-- |
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| + | $#C+1 = 23 : ~/encyclopedia/old_files/data/H110/H.1100140 Heinz\ANDKato inequality | ||
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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 $. | ||
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 | 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 | ||
| − | + | $$ | |
| + | \left ( | ||
| + | |||
| + | \begin{array}{cc} | ||
| + | \left ( {\left | T \right | ^ {2 \alpha } x } , x \right ) &\left ( {Tx } , y \right ) \\ | ||
| + | \left ( y , {Tx } \right ) &\left ( {\left | {T ^ {*} } \right | ^ {2 ( 1 - \alpha ) } y } , y \right ) \\ | ||
| + | \end{array} | ||
| + | |||
| + | \right ) = | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | = | ||
| + | \left ( | ||
| − | + | \begin{array}{cc} | |
| + | \left ( {\left | T \right | ^ \alpha x } , {\left | T \right | ^ \alpha x } \right ) &\left ( {\left | T \right | ^ \alpha x } , {\left | T \right | ^ {1 - \alpha } U ^ {*} y } \right ) \\ | ||
| + | \left ( {\left | T \right | ^ {1 - \alpha } U ^ {*} y } , {\left | T \right | ^ \alpha x } \right ) &\left ( {\left | T \right | ^ {1 - \alpha } U ^ {*} y } , {\left | T \right | ^ {1 - \alpha } U ^ {*} y } \right ) \\ | ||
| + | \end{array} | ||
| + | |||
| + | \right ) | ||
| + | $$ | ||
| − | is non-negative, where | + | is non-negative, where $ T = U | T | $ |
| + | is the [[Polar decomposition|polar decomposition]] of $ T $. | ||
| − | The Heinz–Kato inequality (1952; cf. [[#References|[a4]]], [[#References|[a3]]]): If | + | The Heinz–Kato inequality (1952; cf. [[#References|[a4]]], [[#References|[a3]]]): If $ A $ |
| + | and $ B $ | ||
| + | are positive operators such that $ \| {Tx } \| \leq \| {Ax } \| $ | ||
| + | and $ \| {T ^ {*} y } \| \leq \| {By } \| $ | ||
| + | for all $ x, y \in H $, | ||
| + | then the following inequality holds for all $ x,y \in H $: | ||
| − | + | $$ \tag{a1 } | |
| + | \left | {\left ( {Tx } , y \right ) } \right | \leq \left \| {A ^ \alpha x } \right \| \left \| {B ^ {1 - \alpha } y } \right \| | ||
| + | $$ | ||
| − | for all | + | for all $ \alpha \in [ 0,1 ] $. |
It is proved in [[#References|[a1]]] that the Heinz–Kato inequality is equivalent to: | It is proved in [[#References|[a1]]] that the Heinz–Kato inequality is equivalent to: | ||
| − | + | $$ \tag{a2 } | |
| + | \left \| {A ^ {2} Q } \right \| \geq \left \| {AQA } \right \| | ||
| + | $$ | ||
| − | for arbitrary positive operators | + | for arbitrary positive operators $ A $ |
| + | and $ Q $. | ||
The [[Heinz inequality|Heinz inequality]] yields the Heinz–Kato inequality. | The [[Heinz inequality|Heinz inequality]] yields the Heinz–Kato inequality. | ||
| Line 25: | Line 71: | ||
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: | 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: | ||
| − | + | $$ \tag{a3 } | |
| + | \left \| {A ^ {s} B ^ {s} } \right \| \leq \left \| {AB } \right \| ^ {s} | ||
| + | $$ | ||
| − | for | + | for $ A,B \geq 0 $ |
| + | and $ 0 \leq s \leq 1 $. | ||
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. | 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. | ||
Latest revision as of 22:10, 5 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 $.
The Heinz–Kato inequality is an extension of the generalized Cauchy–Schwarz inequality (cf. also Cauchy inequality). It follows from the fact that
$$ \left ( \begin{array}{cc} \left ( {\left | T \right | ^ {2 \alpha } x } , x \right ) &\left ( {Tx } , y \right ) \\ \left ( y , {Tx } \right ) &\left ( {\left | {T ^ {*} } \right | ^ {2 ( 1 - \alpha ) } y } , y \right ) \\ \end{array} \right ) = $$
$$ = \left ( \begin{array}{cc} \left ( {\left | T \right | ^ \alpha x } , {\left | T \right | ^ \alpha x } \right ) &\left ( {\left | T \right | ^ \alpha x } , {\left | T \right | ^ {1 - \alpha } U ^ {*} y } \right ) \\ \left ( {\left | T \right | ^ {1 - \alpha } U ^ {*} y } , {\left | T \right | ^ \alpha x } \right ) &\left ( {\left | T \right | ^ {1 - \alpha } U ^ {*} y } , {\left | T \right | ^ {1 - \alpha } U ^ {*} y } \right ) \\ \end{array} \right ) $$
is non-negative, where $ T = U | T | $ is the polar decomposition of $ T $.
The Heinz–Kato inequality (1952; cf. [a4], [a3]): If $ A $ and $ B $ are positive operators such that $ \| {Tx } \| \leq \| {Ax } \| $ and $ \| {T ^ {*} y } \| \leq \| {By } \| $ for all $ x, y \in H $, then the following inequality holds for all $ x,y \in H $:
$$ \tag{a1 } \left | {\left ( {Tx } , y \right ) } \right | \leq \left \| {A ^ \alpha x } \right \| \left \| {B ^ {1 - \alpha } y } \right \| $$
for all $ \alpha \in [ 0,1 ] $.
It is proved in [a1] that the Heinz–Kato inequality is equivalent to:
$$ \tag{a2 } \left \| {A ^ {2} Q } \right \| \geq \left \| {AQA } \right \| $$
for arbitrary positive operators $ A $ and $ Q $.
The Heinz inequality yields the Heinz–Kato inequality.
On the other hand, it is shown in [a2] that the 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:
$$ \tag{a3 } \left \| {A ^ {s} B ^ {s} } \right \| \leq \left \| {AB } \right \| ^ {s} $$
for $ A,B \geq 0 $ and $ 0 \leq s \leq 1 $.
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.
References
| [a1] | M. Fujii, T. Furuta, "Löwner–Heinz, Cordes and Heinz–Kato inequalities" Math. Japon. , 38 (1993) pp. 73–78 |
| [a2] | T. Furuta, "Norm inequalities equivalent to Löwner–Heinz theorem" Rev. Math. Phys. , 1 (1989) pp. 135–137 |
| [a3] | T. Kato, "Notes on some inequalities for linear operators" Math. Ann. , 125 (1952) pp. 208–212 |
| [a4] | E. Heinz, "Beiträge zur Störungstheorie der Spektralzerlegung" Math. Ann. , 123 (1951) pp. 415–438 |
Heinz-Kato inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Heinz-Kato_inequality&oldid=16896