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Difference between revisions of "Analytic operator"

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''at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123601.png" />''
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An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123602.png" />, acting from one Banach space into another, that admits a representation of the form
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''at a point $x_0$''
  
<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/a/a012/a012360/a0123603.png" /></td> </tr></table>
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An operator $A$, acting from one [[Banach_space|Banach space]] into another, that admits a representation of the form
 
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\begin{equation}
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123604.png" /> is a form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123605.png" /> and the series converges uniformly in some ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123606.png" />. An operator is called analytic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123607.png" /> if it is an analytic operator at all points of this domain. An analytic operator is infinitely differentiable. In the case of complex spaces, analyticity of an operator in a domain is a consequence of its differentiability (according to Gâteaux) at each point of this domain. Examples of analytic operators are Lyapunov's integro-power series, and the Hammerstein and Urysohn operators with smooth kernels on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012360/a0123608.png" /> of continuous functions.
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A(x_0+h) - Ax_0 = \sum_{k=1}^{\infty}C_kh^k,
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\end{equation}
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where $C_k$ is a form of degree $k$ and the series converges uniformly in some ball $\|h\|<r$. An operator is called analytic in a domain $G$ if it is an analytic operator at all points of this domain. An analytic operator is infinitely differentiable. In the case of complex spaces, analyticity of an operator in a domain is a consequence of its differentiability (according to Gâteaux) at each point of this domain. Examples of analytic operators are Lyapunov's integro-power series, and the Hammerstein and Urysohn operators with smooth kernels on the space $\mathcal C$ of continuous functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,  G.M. Vainikko,  P.P. Zabreiko,  et al.,  "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,  G.M. Vainikko,  P.P. Zabreiko,  et al.,  "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR></table>

Latest revision as of 06:35, 18 December 2012


at a point $x_0$

An operator $A$, acting from one Banach space into another, that admits a representation of the form \begin{equation} A(x_0+h) - Ax_0 = \sum_{k=1}^{\infty}C_kh^k, \end{equation} where $C_k$ is a form of degree $k$ and the series converges uniformly in some ball $\|h\|<r$. An operator is called analytic in a domain $G$ if it is an analytic operator at all points of this domain. An analytic operator is infinitely differentiable. In the case of complex spaces, analyticity of an operator in a domain is a consequence of its differentiability (according to Gâteaux) at each point of this domain. Examples of analytic operators are Lyapunov's integro-power series, and the Hammerstein and Urysohn operators with smooth kernels on the space $\mathcal C$ of continuous functions.

References

[1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian)
How to Cite This Entry:
Analytic operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_operator&oldid=16749
This article was adapted from an original article by P.P. Zabreiko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article