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> | |||
:<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}} | |||
Revision as of 11:54, 7 January 2014
In operator theory, a set is said to be a spectral set for a (possibly unbounded) linear operator on a Banach space if the spectrum of is in and von-Neumann's inequality holds for on - i.e. for all rational functions with no poles on
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.