Compact Operator

The origin of the theory of compact operators is in the theory of Integral Equation s. A typical Fredholm Integral Equation gives rise to a compact operator ''K'' on Function Space s; the compactness property is shown by Equicontinuity . The method of approximation by finite rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm Operator is derived from this connection.

The Spectral Theory for compact operators in the abstract was worked out by Frigyes Riesz (published 1918 ). It shows that a compact operator ''K'' on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or a countably-infinite subset of C which has 0 as its only limit point. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of ''K'' with finite multiplicities (so that ''K'' − λ''I'' has a finite-dimensional Kernel for all complex λ ≠ 0).

The compact operators from a Banach space to itself form a two-sided Ideal in the Algebra of all bounded operators on the space. Indeed, the compact operators on a Hilbert space form a Maximal Ideal , so the Quotient Algebra , known as the Calkin Algebra , is Simple .

Examples of compact operators include Hilbert-Schmidt Operator s, or more generally, operators in the Schmidt Class .

COMPACT OPERATOR ON HILBERT SPACES

An equivalent definition of compact operators on a Hilbert space may be given as follows.

An operator

\mathcal{L}

on a Hilbert Space

\mathcal{H}

\mathcal{L}:\mathcal{H} 	o \mathcal{H}

is said to be ''compact'' if it can be written in the form

:

\mathcal{L} = \sum_{n=1}^N \lambda_n \langle f_n, \cdot 
angle g_n

where

1 \le N \le \infty

and

f_1,\ldots,f_N

and

g_1,\ldots,g_N

are (not necessarily complete) orthonormal sets. Here,

\lambda_1,\ldots,\lambda_N

is a sequence of real or complex numbers, the Singular Value s of the operator, which tends to zero if the sequence is infinite. The bracket

\langle\cdot,\cdot
angle

is the scalar product on the Hilbert space; the sum on the right hand side must converge in the norm.

An important subclass of compact operators are the trace-class or Nuclear Operator s.

SOME PROPERTIES OF COMPACT OPERATORS

In the following, X,Y,Z,W are Banach spaces, B(X,Y) is space of bounded operators from X to Y, K(X,Y) is space of compact operators from X to Y, B(X)=B(X,X), K(X)=K(X,X),

B_X

is the unit ball in X,

id_X

is the Identity Operator on X.

A Bounded Operator $T\in B(X,Y)$ is compact if and only if any of the following is true

--- $T(B_X)$ is Relatively Compact in Y.

--- Image of any bounded set under T is relatively compact in Y.

--- Image of any bounded set under T is Totally Bounded in Y.

--- there exists a Neighbourhood of 0, $U\subset X$ , and compact set $V\subset Y$ such that $T(U)\subset V$ .

--- For any sequence $(x_ n)_{n\in \mathbb N}$ from the unit ball $B_X$ , the sequence $(Tx_n)_{n\in\mathbb N}$ contains a Cauchy Subsequence .

K(X,Y) is closed subspace of B(X,Y)

$B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z)$ This is a generalization of the statement that K(X) forms a two-sided operator ideal in B(X)

$id_X$ is compact if and only if X has finite dimension

For any compact operator $T\in K(X)$ , $id_X-T$ is Fredholm Operator with index 0.

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Information About

Compact Operator