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An integral playing a key role in the Riesz–Dunford functional calculus for Banach spaces (cf. [[Functional calculus|Functional calculus]].) In this calculus, for a fixed bounded [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203101.png" /> on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203102.png" />, all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203103.png" /> holomorphic on a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203104.png" /> of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203106.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]) are turned into a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203107.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d1203108.png" /> by
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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/d/d120/d120310/d1203109.png" /></td> </tr></table>
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This integral is called the Dunford integral. It is assumed here that the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031011.png" /> consists of a finite number of rectifiable Jordan curves (cf. also [[Jordan curve|Jordan curve]]), oriented in positive sense.
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An integral playing a key role in the Riesz–Dunford functional calculus for Banach spaces (cf. [[Functional calculus|Functional calculus]].) In this calculus, for a fixed bounded [[Linear operator|linear operator]] $T$ on a [[Banach space|Banach space]] $X$, all functions $f$ holomorphic on a neighbourhood $U$ of the spectrum $\sigma ( T )$ of $T$ (cf. also [[Spectrum of an operator|Spectrum of an operator]]) are turned into a bounded linear operator $f ( T )$ on $X$ by
  
For suitably chosen domains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031013.png" />, the following rules of operational calculus hold:
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\begin{equation*} f ( T ) = \frac { 1 } { 2 \pi i } \int _ { \partial U } f ( \lambda ) ( \lambda - T ) ^ { - 1 } d \lambda. \end{equation*}
  
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This integral is called the Dunford integral. It is assumed here that the boundary $\partial U$ of $U$ consists of a finite number of rectifiable Jordan curves (cf. also [[Jordan curve|Jordan curve]]), oriented in positive sense.
  
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For suitably chosen domains of $f$ and $g$, the following rules of operational calculus hold:
  
Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031017.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031018.png" /> in the operator norm. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120310/d12031020.png" />.
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\begin{equation*} \alpha f ( T ) + \beta g ( T ) = ( \alpha f + \beta g ) ( T ), \end{equation*}
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\begin{equation*} f ( T ) g ( T ) = ( f g ) ( T ) , f ( \sigma ( T ) ) = \sigma ( f ( T ) ). \end{equation*}
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Also, $f ( \lambda ) = \sum _ { n = 0 } ^ { \infty } \alpha _ { n } \lambda ^ { n }$ on $U$ implies $f ( T ) = \sum _ { n = 0 } ^ { \infty } \alpha _ { n } T ^ { n }$ in the operator norm. If $h ( \lambda ) = g ( f ( \lambda ) )$, then $h ( T ) = g ( f ( T ) )$.
  
 
The Dunford integral can be considered as a [[Bochner integral|Bochner integral]].
 
The Dunford integral can be considered as a [[Bochner integral|Bochner integral]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators" , '''1''' , Interscience  (1958)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)</td></tr></table>

Latest revision as of 17:00, 1 July 2020

An integral playing a key role in the Riesz–Dunford functional calculus for Banach spaces (cf. Functional calculus.) In this calculus, for a fixed bounded linear operator $T$ on a Banach space $X$, all functions $f$ holomorphic on a neighbourhood $U$ of the spectrum $\sigma ( T )$ of $T$ (cf. also Spectrum of an operator) are turned into a bounded linear operator $f ( T )$ on $X$ by

\begin{equation*} f ( T ) = \frac { 1 } { 2 \pi i } \int _ { \partial U } f ( \lambda ) ( \lambda - T ) ^ { - 1 } d \lambda. \end{equation*}

This integral is called the Dunford integral. It is assumed here that the boundary $\partial U$ of $U$ consists of a finite number of rectifiable Jordan curves (cf. also Jordan curve), oriented in positive sense.

For suitably chosen domains of $f$ and $g$, the following rules of operational calculus hold:

\begin{equation*} \alpha f ( T ) + \beta g ( T ) = ( \alpha f + \beta g ) ( T ), \end{equation*}

\begin{equation*} f ( T ) g ( T ) = ( f g ) ( T ) , f ( \sigma ( T ) ) = \sigma ( f ( T ) ). \end{equation*}

Also, $f ( \lambda ) = \sum _ { n = 0 } ^ { \infty } \alpha _ { n } \lambda ^ { n }$ on $U$ implies $f ( T ) = \sum _ { n = 0 } ^ { \infty } \alpha _ { n } T ^ { n }$ in the operator norm. If $h ( \lambda ) = g ( f ( \lambda ) )$, then $h ( T ) = g ( f ( T ) )$.

The Dunford integral can be considered as a Bochner integral.

References

[a1] N. Dunford, J.T. Schwartz, "Linear operators" , 1 , Interscience (1958)
[a2] K. Yosida, "Functional analysis" , Springer (1980)
How to Cite This Entry:
Dunford integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dunford_integral&oldid=50341
This article was adapted from an original article by J. de Graaf (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article