Editing Compact operator (section)

== Properties ==

In the following, <math>X, Y, Z, W</math> are Banach spaces, <math>B(X,Y)</math> is the space of bounded operators <math>X \to Y</math> under the [[operator norm]], and <math>K(X,Y)</math> denotes the space of compact operators <math>X \to Y</math>. <math>\operatorname{Id}_X</math> denotes the [[identity operator]] on <math>X</math>, <math>B(X) = B(X,X)</math>, and <math>K(X) = K(X,X)</math>.

* <math>K(X,Y)</math> is a closed subspace of <math>B(X,Y)</math> (in the norm topology). Equivalently,{{sfn | Rudin | 1991 | pp=103-115}}  
** given a sequence of compact operators  <math>(T_n)_{n \in \mathbf{N}}</math> mapping <math>X \to Y</math> (where <math>X,Y</math>are Banach) and given that <math>(T_n)_{n \in \mathbf{N}}</math> converges to <math>T</math> with respect to the [[operator norm]], ''<math>T</math>'' is then compact.
* Conversely, if <math>X,Y</math> are Hilbert spaces, then every compact operator from <math>X \to Y</math> is the limit of finite rank operators. Notably, this "[[approximation property]]" is false for general Banach spaces ''X'' and ''Y''.<ref name=":0" />
*<math>B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z),</math> where the [[function composition|composition]] of sets is taken element-wise. In particular, <math>K(X)</math> forms a two-sided [[ideal (ring theory)|ideal]] in <math>B(X)</math>.
*Any compact operator is [[strictly singular]], but not vice versa.<ref>N.L. Carothers, ''A Short Course on Banach Space Theory'', (2005) London Mathematical Society Student Texts '''64''', Cambridge University Press.</ref>
* A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (''Schauder's theorem'').{{sfn | Conway | 1990 | pp=173-177}} 
** If <math>T: X \to Y</math> is bounded and compact, then:{{sfn | Rudin | 1991 | pp=103-115}}{{sfn | Conway | 1990 | pp=173-177}}
*** the closure of the range of ''<math>T</math>'' is [[Separable space|separable]].
*** if the range of ''<math>T</math>'' is closed in ''Y'', then the range of ''<math>T</math>'' is finite-dimensional.
* If <math>X</math> is a Banach space and there exists an [[invertible]] bounded compact operator <math>T: X \to X</math> then ''<math>X</math>'' is necessarily finite-dimensional.{{sfn|Conway|1990|pp=173-177}}

Now suppose that <math>X</math> is a Banach space and <math>T\colon X \to X</math> is a compact linear operator, and <math>T^* \colon X^* \to X^*</math> is the [[Hermitian adjoint|adjoint]] or [[transpose]] of ''T''.
* For any <math>T\in K(X)</math>, <math>{\operatorname{Id}_X} - T</math>&thinsp; is a [[Fredholm operator]] of index 0.  In particular, <math>\operatorname{Im}({\operatorname{Id}_X} - T)</math> is closed. This is essential in developing the spectral properties of compact operators.  One can notice the similarity between this property and the fact that, if ''<math>M</math>'' and ''<math>N</math>'' are subspaces of ''<math>X</math>'' where <math>M</math> is closed and ''<math>N</math>'' is finite-dimensional, then <math>M+N</math> is also closed.
* If <math>S\colon X \to X</math> is any bounded linear operator then both <math>S \circ T</math> and <math>T \circ S</math> are compact operators.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>\lambda \neq 0</math> then the range of <math>T - \lambda \operatorname{Id}_X</math> is closed and the kernel of <math>T - \lambda \operatorname{Id}_X</math> is finite-dimensional.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>\lambda \neq 0</math> then the following are finite and equal: <math>\dim \ker \left( T - \lambda \operatorname{Id}_X \right) 
= \dim\big(X / \operatorname{Im}\left( T - \lambda \operatorname{Id}_X \right) \big)
= \dim \ker \left( T^* - \lambda \operatorname{Id}_{X^*} \right)
= \dim\big(X^* / \operatorname{Im}\left( T^* - \lambda \operatorname{Id}_{X^*} \right) \big)</math>{{sfn | Rudin | 1991 | pp=103-115}}
* The [[Spectrum (functional analysis)|spectrum]] <math>\sigma(T)</math> of ''<math>T</math>'' is compact, [[countable]], and has at most one [[limit point]], which would necessarily be the origin.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>X</math> is infinite-dimensional then <math>0 \in \sigma(T)</math>.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>\lambda \neq 0</math> and <math>\lambda \in \sigma(T)</math> then <math>\lambda</math> is an eigenvalue of both ''<math>T</math>'' and <math>T^{*}</math>.{{sfn | Rudin | 1991 | pp=103-115}}
* For every <math>r > 0</math> the set <math>E_r = \left\{ \lambda \in \sigma(T) : | \lambda | > r \right\}</math> is finite, and for every non-zero <math>\lambda \in \sigma(T)</math> the range of <math>T - \lambda \operatorname{Id}_X</math> is a [[proper subset]] of ''X''.{{sfn | Rudin | 1991 | pp=103-115}}