Stochastic approximation: Difference between revisions

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In [[operator theory]], a set <math>X\subseteq\mathbb{C}</math> is said to be a '''spectral set''' for a (possibly unbounded) linear operator <math>T</math> on a Banach space if the [[Spectrum of an operator|spectrum]] of <math>T</math> is in <math>X</math> and von-Neumann's inequality holds for <math>T</math> on <math>X</math> - i.e. for all rational functions <math>r(x)</math> with no [[pole (complex analysis)|poles]] on <math>X</math>
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:<math>\left\Vert r(T) \right\Vert \leq \left\Vert r \right\Vert_{X} = \sup \left\{\left\vert r(x) \right\vert : x\in X \right\}</math>
 
This concept is related to the topic of analytic functional calculus
of operators. In general, one wants to get more details about the operators constructed from functions with the original operator as the variable.
 
{{DEFAULTSORT:Spectral Set}}
[[Category:Functional analysis]]
 
{{Mathanalysis-stub}}

Latest revision as of 18:55, 27 December 2014

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