Compact operator

Compact operators between two Banach spaces are in functional analysis, one of the branches of mathematics, special operators, which have their origin in the theory of integral equations. This is also called compact pictures instead of compact operators and different linear of nonlinear operators.

• 2.1 Definition
• 2.2 Approximation by operators with nite dimensional
• 2.3 fixed point theory

Theory of linear compact operators

Definition

A linear mapping of a Banach space into a Banach space is called compact operator if one of the following equivalent properties is satisfied:

• The operator maps every bounded subset of a relatively compact subset of.
• The image of the open ( or closed ) unit ball in is relatively compact in.
• Every bounded sequence in has a subsequence such that converges.

The amount of the linear compact operators is referred to herein.

Continuity

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

Examples

• A linear operator of finite rank, ie an operator with nite dimensional compact.
• Hilbert - Schmidt operators and trace class operators are always compact.
• The identity on a Banach space is compact if the Banach space is finite. This follows from the fact that the unit ball is relatively compact if and only if the Banach space is finite. Comparisons to compactness theorem of Riesz.

Properties

• Is complete, so is also a Banach space. That is, for a compact and a scalar operators are operators and compact. In addition, every Cauchy sequence converges with respect to the operator norm to a linear compact operator.
• The linear operator is compact if every bounded sequence exists in a subsequence of which converges in. So compact operators form from bounded sequences to sequences with convergent subsequences. Is infinite-dimensional, there is limited impact having no convergent consequences. Thus compact operators 'improve' convergence characteristics.
• Be, and normed spaces, a compact operator, and bounded operators. Then is also compact.
• In particular, the set of all compact operators of a Hilbert space is a selbstadjungiertes closed ideal in the C *-algebra of all bounded linear operators on.

Set of Schauder

The following theorem is named after Juliusz Schauder. Let and be Banach spaces. Then a linear operator is compact if the adjoint operator is compact.

Approximation property

If a linear operator between the Banach spaces and and there exists a sequence of continuous linear operators with nite dimensional, converging to, so is compact. The converse is not true in general, but only if the so-called approximation property has. However, many of Banach spaces have frequently used this approximation property, for example, or with, and all Hilbert spaces.

Spectral theory of compact operators on Banach spaces

Be a Banach space and a compact operator. With the spectrum of the operator is called. If the room is also infinite dimensional, so true and the possibly empty set has at most countably many elements. In particular, the only possible accumulation point is.

Each is an eigenvalue of and the corresponding eigenspace is finite-dimensional. In addition, a topological direct decomposition exists with and, where is finite and covers, as well as an isomorphism from to is. This decomposition is called Riesz decomposition and is named after the mathematician Frigyes Riesz, large parts of the spectral theory of ( compact ) has explored operators.

Spectral decomposition of self-adjoint compact operators on Hilbert spaces

Is a compact self-adjoint operator on a Hilbert space, then there exists a spectral decomposition for the operator. This means that there exists an orthonormal system and a null sequence such that

Applies to all. These are for all the eigenvalues ​​of and is an eigenvector.

General spectral decomposition of compact operators on Hilbert spaces

More generally, a compact operator on the Hilbert spaces and then you can the above result to the two operators, and apply (this is for an operator, the amount is a positive (and hence self-adjoint ) operator, is for the; , this operator always exists and it is unique ).

One then obtains orthonormal systems and as well as a null sequence such that

And

Applies to all.

Similarly as above then are the eigenvalues ​​of and, and the eigenvectors of the eigenvectors.

Application

Be compact with real positive Lebesgue measure and steadily on. Then by

Defined Fredholm integral operator is a linear compact operator. This statement can be proved by means of the set of Arzelà - Ascoli.

Many sentences for solvability of integral equations, such as the Fredholm alternative, require a compact operator.

Schmidt- representation and the shadow class

Let and be Hilbert spaces and a compact operator. Then there exist a countable orthonormal systems and as well as numbers, so that

Applies to all. This representation of the compact operator is called the Schmidt- representation and the numbers are in contrast to the orthonormal systems uniquely determined and are called singular numbers. Valid for it is said, that is the p-th shadow class. If so, the operators of nuclear and hot, so it is a Hilbert-Schmidt operator. On the amount of the Hilbert - Schmidt operators can be defined, in contrast to the other shadow classes naturally a Hilbert space structure.

Full Continuous Operators

Let and be Banach spaces, operator. Then is called completely continuous if for each image sequence is weakly convergent sequence in normkonvergent. Compact operators are completely continuous. Is reflexive, so every completely continuous operator is compact.

Nonlinear Compact Operators

Definition

Let and be normed spaces, an operator. Then is compact if is continuous and the image of each bounded set into a relatively compact subset of is. The amount of compact operators is referred to herein.

Approximation by operators with nite dimensional

Let and be normed spaces and a bounded closed subset. With the space of compact operators whose image is contained in a finite dimensional subspace of, respectively. Be a compact operator, then there exists for every a compact operator, so that

Applies. This means that the room is located with respect to the supremum norm dense in the space of compact operators. If a Banach space, then the converse also holds. That is, a sequence of operators from which converges with respect to the supremum norm, has as limit a compact operator. So the space of compact operators with limited is particularly complete.

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

Fixed point theory

Many nonlinear differential and integral equations can be written as a short equation, where a compact operator. For such nonlinear problems, there is no comprehensive solution theory. A possibility to examine the equation solutions, the fixed point theory. In this context, for example, the fixed point theorem of Schauder or Leray - Schauder alternative central tools that guarantee the existence of fixed points. In addition, it can be shown that if is closed and bounded, the set of fixed points of a compact operator is compact.