Compact Operators

Definition (Compact Linear Operator)

A compact linear operator between two normed spaces is a linear operator that maps bounded sets to relatively compact sets.

In other words, if $X$ and $Y$ are normed spaces, and $T:X\to Y$ is a linear operator, then we call $T$ compact, if for every bounded subset $M\subset X$ the image $TM$ is relatively compact in $Y$, meaning its closure is compact.

All compact operators are bounded.

Proposition (Characterization of Compactness)

A linear operator between two normed spaces is compact if and only if it maps any bounded sequence into one that contains a convergent subsequence.

Let $X$ and $Y$ be normed spaces. A linear operator $T:X\to Y$ is compact if and only if for every bounded sequence $(x_n)$ in $X$ the image sequence $(T{x}_n)$ in $Y$ has a convergent subsequence.

Proposition (Compactness of Zero and Identity)

The zero operator on any normed space is compact. The identity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.

Proposition (The Space of Compact Linear Operators)

The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$ form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$. If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space $B(X,Y)$ and hence itself a Banach space.

examples:

• integral operators
• finite-rank operators
• Hilbert-Schmidt operators
• …