compact operator

Let $X$ and $Y$ be two Banach spaces.A compact operator (completely continuous operator) is a linear operator $T:X\to Y$that maps the unit ball in $X$ to a set in $Y$ with compact closure. It can be shown that a compact operator is necessarily a bounded operator.

The set of all compact operators on $X$, commonly denoted by $𝕂(X)$,is a closed two-sided ideal of the set of all bounded operators on $X$, $𝔹(X)$.

Any bounded operator which is the norm limit of a sequence of finite rank operators is compact.In the case of Hilbert spaces, the converse is also true.That is, any compact operator on a Hilbert space is a norm limit of finite rank operators.

Example 1 (Integral operators)

Let $k\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}$, with $x\mathrm{,}y\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{]}$, be a continuous function.The operator defined by

$$(T\psi )(x)={\int }_0^{1}k(x,y)\psi (y)dy,\psi \in C([0,1])$$

is compact.