Compact operator

In functional analysis , one of the branches of mathematics , compact operators between two Banach spaces are special operators that have their origin in the theory of integral equations . One speaks of compact mappings instead of compact operators and distinguishes between linear and non-linear operators.

Theory of linear compact operators

definition

A linear mapping from a Banach space into a Banach space is called a compact operator if one of the following equivalent properties is fulfilled: ${\displaystyle K\colon E\to F}$ ${\displaystyle E}$ ${\displaystyle F}$

• The operator maps each bounded subset of to a relatively compact subset of . ${\displaystyle K}$ ${\displaystyle E}$ ${\displaystyle F}$
• The image of the open (or closed) unit sphere in is relatively compact in . ${\displaystyle E}$ ${\displaystyle F}$
• Every bounded sequence in has a subsequence so that it converges. ${\displaystyle (x_{n})}$ ${\displaystyle E}$ ${\displaystyle (x_{n_{k}})}$ ${\displaystyle (Kx_{n_{k}})}$

The set of linear, compact operators is denoted here by . ${\displaystyle K\colon E\to F}$ ${\displaystyle {\mathcal {K}}(E,F)}$

continuity

Because the image of the unit sphere is relatively compact and thus bounded, it follows that every linear compact operator is automatically a bounded operator and is therefore continuous.

Examples

• A continuous linear operator of finite rank, that is, an operator with a finite-dimensional image, is compact.
• Hilbert-Schmidt operators and track class operators are always compact.
• The identity on a Banach space is compact if and only if the Banach space is finite-dimensional. This follows from the fact that the unit sphere is relatively compact if and only if the Banach space is finite-dimensional. Compare Riesz's compactness theorem .

properties

• If complete, then there is also a Banach space. That is, for compact operators and a scalar , the operators and are compact. Furthermore, every Cauchy sequence converges with respect to the operator norm to a linear compact operator . ${\displaystyle F}$ ${\displaystyle {\mathcal {K}}(E,F)}$ ${\displaystyle K_{1},K_{2}}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle K_{1}+K_{2}}$ ${\displaystyle \lambda K}$ ${\displaystyle (K_{n})_{n=1}^{\infty }}$ ${\displaystyle \textstyle \lim _{n\to \infty }K_{n}}$
• The linear operator is compact if and only if for every bounded sequence in there exists a subsequence of which converges in. Compact operators therefore map restricted sequences to sequences with convergent partial sequences. Is infinite dimensional, there are bounded sequences that have no convergent subsequences. Thus, compact operators can "improve" convergence properties. ${\displaystyle K\colon E\to F}$ ${\displaystyle (x_{n})}$ ${\displaystyle E}$ ${\displaystyle (K(x_{n}))}$ ${\displaystyle F}$ ${\displaystyle E}$
• Be , , and normed spaces, a compact operator, and bounded operators. Then it is also compact. ${\displaystyle W}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle K:X\rightarrow Y}$ ${\displaystyle A:W\rightarrow X}$ ${\displaystyle B:Y\rightarrow Z}$ ${\displaystyle BKA:W\rightarrow Z}$
• In particular, the set of all compact operators of a Hilbert space is a self-adjoint closed ideal in the C * -algebra of all bounded linear operators on . ${\displaystyle H}$ ${\displaystyle H}$

Theorem of shudders

The following sentence is named after Juliusz Schauder . Be and Banach spaces. Then a linear operator is compact if and only if the adjoint operator is compact. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle K\colon X\to Y}$ ${\displaystyle K^{*}\colon Y^{*}\to X^{*}}$

Approximation property

If there is a linear operator between the Banach spaces and and if there is a sequence of continuous linear operators with a finite-dimensional image that converges to, then is compact. In general, the converse does not apply, but only if it has the so-called approximation property . Many of the frequently used Banach spaces have this approximation property, for example , or with , as well as all Hilbert spaces . ${\displaystyle K\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle Y}$ ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}([0,1])}$ ${\displaystyle 1\leq p<\infty }$

Spectral theory of compact operators on Banach spaces

Let be a Banach space and a compact operator. With that is spectrum of the operator referred. If the space is also infinitely dimensional, it holds that the possibly empty set has at most a countable number of elements. In particular, the only possible accumulation point of . ${\displaystyle X}$ ${\displaystyle T\colon X\to X}$ ${\displaystyle \sigma (T)}$ ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle 0\in \sigma (T)}$ ${\displaystyle \sigma (T)\setminus \{0\}}$ ${\displaystyle 0}$ ${\displaystyle \sigma (T)}$

Each is an eigenvalue of and the associated eigenspace is finite dimensional. In addition, a exists topologically direct decomposition of and , wherein is finite and comprises, as well as an isomorphism of to is. This decomposition is called Riesz decomposition and is named after the mathematician Frigyes Riesz , who researched large parts of the spectrum theory of (compact) operators. ${\displaystyle \lambda \in \sigma (T)\setminus \{0\}}$ ${\displaystyle T}$ ${\displaystyle \operatorname {ker} (\lambda \operatorname {Id} -T)}$ ${\displaystyle X=N(\lambda )\oplus R(\lambda )}$ ${\displaystyle T(N(\lambda ))\subset N(\lambda )}$ ${\displaystyle T(R(\lambda ))\subset R(\lambda )}$ ${\displaystyle N(\lambda )}$ ${\displaystyle \operatorname {ker} (\lambda \operatorname {Id} -T)}$ ${\displaystyle (\lambda \operatorname {Id} -T)|_{R(\lambda )}}$ ${\displaystyle R(\lambda )}$ ${\displaystyle R(\lambda )}$

Spectral decomposition of normal compact operators on Hilbert spaces

If a compact normal operator is on a Hilbert space , then there is a spectral decomposition for the operator . That is, there is an orthonormal system as well as a null sequence in such that ${\displaystyle T\colon H\to H}$ ${\displaystyle H}$ ${\displaystyle e_{1},e_{2},\ldots }$ ${\displaystyle (\lambda _{k})_{k\in \mathbb {N} }}$ ${\displaystyle \mathbb {K} \backslash \{0\}}$

${\displaystyle Tx=\sum _{k=1}^{\infty }\lambda _{k}\langle x,e_{k}\rangle e_{k}}$

applies to all . They are for all the eigenvalues of and is an eigenvector too . ${\displaystyle x\in H}$ ${\displaystyle \lambda _{k}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle T}$ ${\displaystyle e_{k}}$ ${\displaystyle \lambda _{k}}$

If it is also self-adjoint, that is , then all eigenvalues ​​are real. If is also positive, i.e. for all , then all eigenvalues ​​are positive real. ${\displaystyle T}$ ${\displaystyle T=T^{*}}$ ${\displaystyle T}$ ${\displaystyle \langle Tx,x\rangle \geq 0}$ ${\displaystyle x\in H}$

Spectral decomposition of general compact operators on Hilbert spaces

Is more generally a compact operator on the Hilbert spaces and then the above result can be applied to the two operators and (where the absolute value for an operator is a positive (and therefore self-adjoint) operator for which is; this operator always exists and it is unique ). ${\displaystyle T\colon H_{1}\to H_{2}}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle |T|\colon H_{1}\to H_{1}}$ ${\displaystyle |T^{*}|\colon H_{2}\to H_{2}}$ ${\displaystyle A}$ ${\displaystyle |A|}$ ${\displaystyle |A|^{2}=A^{*}A}$

One then obtains orthonormal systems of and of as well as a zero sequence in such that ${\displaystyle e_{1},e_{2},\ldots }$ ${\displaystyle H_{1}}$ ${\displaystyle f_{1},f_{2},\ldots }$ ${\displaystyle H_{2}}$ ${\displaystyle (\lambda _{k})_{k\in \mathbb {N} }}$ ${\displaystyle \mathbb {K} \backslash \{0\}}$

${\displaystyle Tx=\sum _{k=1}^{\infty }\lambda _{k}\langle x,e_{k}\rangle f_{k}}$

${\displaystyle x\in H_{1}}$ and

${\displaystyle T^{*}y=\sum _{k=1}^{\infty }\lambda _{k}\langle y,f_{k}\rangle e_{k}}$

applies to all . ${\displaystyle y\in H_{2}}$

The eigenvalues ​​of and , the eigenvectors of and the eigenvectors of are then similar to the above . ${\displaystyle \lambda _{k}}$ ${\displaystyle |T|}$ ${\displaystyle |T^{*}|}$ ${\displaystyle e_{k}}$ ${\displaystyle |T|}$ ${\displaystyle f_{k}}$ ${\displaystyle |T^{*}|}$

application

Be compact with a really positive Lebesgue measure and steadily on . Then it's through ${\displaystyle G\subseteq \mathbb {R} }$ ${\displaystyle k}$ ${\displaystyle G\times G}$

${\displaystyle Tx(t)=\int \limits _{G}k(t,s)x(s)\mathrm {d} s}$

defined Fredholm's integral operator a linear compact operator. This statement can be proven with the help of Arzelà-Ascoli's theorem.

Many theorems about the solvability of integral equations, such as Fredholm's alternative , require a compact operator.

Schmidt representation and the shadow class

→ Main article : Shadow class

Let and Hilbert spaces and be a compact operator. Then there exist countable orthonormal systems of and of as well as numbers with such that ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle T\colon H_{1}\to H_{2}}$ ${\displaystyle (e_{i})_{i\in \mathbb {N} }}$ ${\displaystyle H_{1}}$ ${\displaystyle (f_{i})_{i\in \mathbb {N} }}$ ${\displaystyle H_{2}}$ ${\displaystyle s_{1}\geq s_{2}\geq \ldots \geq 0}$ ${\displaystyle s_{k}\to 0}$

${\displaystyle Tx=\sum _{k=1}^{\infty }s_{k}\langle x,e_{k}\rangle f_{k}}$

applies to all . This representation of the compact operator is called the Schmidt representation and , in contrast to the orthonormal systems, the numbers are uniquely determined and are called singular numbers. Is true for so one says that lies in the p th shadow class. If the operators are called nuclear and is , then it is a Hilbert-Schmidt operator . In contrast to the other shadow classes, a Hilbert space structure can be defined naturally on the set of Hilbert-Schmidt operators. ${\displaystyle x\in H_{1}}$ ${\displaystyle s_{i}}$ ${\displaystyle (s_{i})_{i\in \mathbb {N} }\in \ell ^{p}}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle T}$ ${\displaystyle p=1}$ ${\displaystyle p=2}$

Complete operators

→ Main article : Complete operator

Be and Banach spaces, an operator. Then is called complete if for each in weakly convergent sequence the image sequence in is norm-convergent. Compact operators are complete. If reflexive, every complete operator is also compact. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle K\colon E\to F}$ ${\displaystyle K}$ ${\displaystyle E}$ ${\displaystyle (x_{n})}$ ${\displaystyle (K(x_{n}))}$ ${\displaystyle F}$ ${\displaystyle E}$

Nonlinear compact operators

definition

Be and standardized spaces , an operator. Then say , compact case is continuous and the image of each bounded set in a relatively compact subset of is. The set of compact operators is denoted here by . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle K\colon \Omega \subset E\to F}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle S}$ ${\displaystyle \Omega }$ ${\displaystyle F}$ ${\displaystyle {\mathcal {R}}(E,F)}$

Note that here the continuity does not follow automatically as in the linear case, but has to be explicitly required.

Approximation by operators with a finite-dimensional image

Let and be standardized spaces and a bounded, closed subset. The space of compact operators , whose image is contained in a finite-dimensional subspace of , is designated by. Let be a compact operator, then for each there is a compact operator such that ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle \Omega \subset E}$ ${\displaystyle {\mathcal {F}}(\Omega ,F)}$ ${\displaystyle L}$ ${\displaystyle L(\Omega )}$ ${\displaystyle F}$ ${\displaystyle K\colon \Omega \to Y}$ ${\displaystyle \epsilon >0}$ ${\displaystyle K_{\epsilon }\in {\mathcal {F}}(\Omega ,F)}$

${\displaystyle \sup _{x\in \Omega }\|K(x)-K_{\epsilon }(x)\|_{F}<\epsilon }$

applies. This means that the space is close to the space of the compact operators with regard to the supremum norm. If a Banach space , then the converse is also true. That is, a sequence of operators that converges with respect to the supremum norm has a compact operator as its limit. So in particular the space of compact operators with bounded is complete . ${\displaystyle {\mathcal {F}}(\Omega ,F)}$ ${\displaystyle \textstyle \sup _{x\in \Omega }\|\cdot \|_{F}}$ ${\displaystyle {\mathcal {R}}(\Omega ,F)}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {F}}(\Omega ,F)}$ ${\displaystyle {\mathcal {R}}(\Omega ,F)}$ ${\displaystyle \Omega }$

Note that an approximation of this kind is always possible and does not require, as in the linear case described above, that the Banach space involved has the approximation property.

Fixed point theory

Many nonlinear differential and integral equations can be briefly written as equations , where is a compact operator. There is no comprehensive solution theory for such nonlinear problems. One way to examine the equation for solutions is to use fixed point theory. In this context, for example, Schauder's fixed point theorem or the Leray-Schauder alternative are central aids that guarantee the existence of fixed points. It can also be shown that if it is closed and bounded, the set of fixed points of a compact operator is compact. ${\displaystyle F(x)=y}$ ${\displaystyle F\colon \Omega \to X}$ ${\displaystyle \Omega \subset X}$

Individual evidence

1. ↑ This is - along with others such as Schauder-Mazur's theorem - one of numerous sentences that can be attributed to Juliusz Schauder.
2. ↑ Dirk Werner : functional analysis , Springer-Verlag, Berlin, 2005, ISBN 3-540-21381-3 , p. 70
3. ^ John B. Conway: A Course in Functional Analysis . 2nd Edition. Springer, ISBN 0-387-97245-5 , VI, §3
4. ^ Klaus Deimling: Nonlinear Functional Analysis. 1 . Edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , page 55.