Difference between revisions of "Spectral theory of compact operators"
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''Riesz theory of compact operators'' | ''Riesz theory of compact operators'' | ||
| − | Let | + | Let $X$ be a complex [[Banach space|Banach space]] and $T$ a [[Compact operator|compact operator]] on $X$. Then $\sigma ( T )$, the spectrum of $T$, is countable and has no cluster points except, possibly, $0$. Every $0 \neq \lambda \in \sigma ( T )$ is an eigenvalue, and a pole of the resolvent function $\lambda \mapsto ( T - \lambda I ) ^ { - 1 }$. Let $\nu ( \lambda )$ be the order of the pole $\lambda$. For each $n \in \mathbf N$, $( T - \lambda I ) ^ { n } X$ is closed, and this range is constant for $n \geq \nu ( \lambda )$. The null space $N ( ( T - \lambda I ) ^ { n } )$ is finite dimensional and constant for $n \geq \nu ( \lambda )$. The spectral projection $E ( \lambda )$ (the Riesz projector, see [[Riesz decomposition theorem|Riesz decomposition theorem]]) has non-zero finite-dimensional range, equal to $N ( ( T - \lambda I ) ^ { \nu ( \lambda ) } )$, and its null space is $( T - \lambda l ) ^ { \nu ( \lambda ) } X$. Finally, $\operatorname { dim } ( E ( \lambda ) X ) \geq \nu ( \lambda ) \geq 1$. |
| − | The respective integers | + | The respective integers $\nu ( \lambda )$ and $\operatorname { dim } ( E ( \lambda ) X )$ are called the index and the algebraic multiplicity of the eigenvalue $\lambda \neq 0$. |
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) pp. 45ff.</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> N. Dunford, J.T. Schwartz, "Linear operators I: General theory" , Interscience (1964) pp. Sect. VII.4</td></tr></table> |
Latest revision as of 17:00, 1 July 2020
Riesz theory of compact operators
Let $X$ be a complex Banach space and $T$ a compact operator on $X$. Then $\sigma ( T )$, the spectrum of $T$, is countable and has no cluster points except, possibly, $0$. Every $0 \neq \lambda \in \sigma ( T )$ is an eigenvalue, and a pole of the resolvent function $\lambda \mapsto ( T - \lambda I ) ^ { - 1 }$. Let $\nu ( \lambda )$ be the order of the pole $\lambda$. For each $n \in \mathbf N$, $( T - \lambda I ) ^ { n } X$ is closed, and this range is constant for $n \geq \nu ( \lambda )$. The null space $N ( ( T - \lambda I ) ^ { n } )$ is finite dimensional and constant for $n \geq \nu ( \lambda )$. The spectral projection $E ( \lambda )$ (the Riesz projector, see Riesz decomposition theorem) has non-zero finite-dimensional range, equal to $N ( ( T - \lambda I ) ^ { \nu ( \lambda ) } )$, and its null space is $( T - \lambda l ) ^ { \nu ( \lambda ) } X$. Finally, $\operatorname { dim } ( E ( \lambda ) X ) \geq \nu ( \lambda ) \geq 1$.
The respective integers $\nu ( \lambda )$ and $\operatorname { dim } ( E ( \lambda ) X )$ are called the index and the algebraic multiplicity of the eigenvalue $\lambda \neq 0$.
References
| [a1] | H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) pp. 45ff. |
| [a2] | N. Dunford, J.T. Schwartz, "Linear operators I: General theory" , Interscience (1964) pp. Sect. VII.4 |
How to Cite This Entry:
Spectral theory of compact operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_theory_of_compact_operators&oldid=18215
Spectral theory of compact operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_theory_of_compact_operators&oldid=18215
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article