Difference between revisions of "Spectral measure"
From Encyclopedia of Mathematics
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| − | A unitary homomorphism from some Boolean algebra of sets into the Boolean algebra of projection operators on a Banach space. Every operator | + | A unitary homomorphism from some [[Boolean algebra]] of sets into the Boolean algebra of projection operators on a [[Banach space]]. Every operator $T$ on a Banach space $X$ defines a spectral measure on the set of [[Open-closed set|open-and-closed subset]]s of its [[Spectrum of an operator|spectrum]] $\sigma(T)$ by the formula |
| + | $$ | ||
| + | E(\alpha) = \frac{1}{2 \pi i} \int_\Gamma (zI-T)^{-1} dz \ , | ||
| + | $$ | ||
| + | where $\Gamma$ is a [[Jordan curve]] separating $\alpha$ from $\sigma(T) \setminus \alpha$. Here, $TE(\alpha) = E(\alpha)T$ and $\sigma\left(T \downharpoonright_{E(\alpha)X}\right) \subseteq \bar\alpha$. The construction of spectral measures satisfying these conditions on wider classes of Boolean algebras of sets is one of the basic problems in the [[spectral theory]] of linear operators. | ||
| − | <table | + | ====References==== |
| + | <table> | ||
| + | <TR><TD valign="top">[1a]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , '''3''' , Interscience (1971)</TD></TR> | ||
| + | <TR><TD valign="top">[1b]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , '''2''' , Interscience (1963)</TD></TR> | ||
| + | </table> | ||
| − | + | {{TEX|done}} | |
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Latest revision as of 18:24, 22 April 2016
A unitary homomorphism from some Boolean algebra of sets into the Boolean algebra of projection operators on a Banach space. Every operator $T$ on a Banach space $X$ defines a spectral measure on the set of open-and-closed subsets of its spectrum $\sigma(T)$ by the formula $$ E(\alpha) = \frac{1}{2 \pi i} \int_\Gamma (zI-T)^{-1} dz \ , $$ where $\Gamma$ is a Jordan curve separating $\alpha$ from $\sigma(T) \setminus \alpha$. Here, $TE(\alpha) = E(\alpha)T$ and $\sigma\left(T \downharpoonright_{E(\alpha)X}\right) \subseteq \bar\alpha$. The construction of spectral measures satisfying these conditions on wider classes of Boolean algebras of sets is one of the basic problems in the spectral theory of linear operators.
References
| [1a] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971) |
| [1b] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) |
How to Cite This Entry:
Spectral measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_measure&oldid=17065
Spectral measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_measure&oldid=17065
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article