# finite rank approximation on separable Hilbert spaces

Theorem Let $\mathscr{H}$ be a separable Hilbert space and let $T\in L(\mathscr{H})$. Then $T$ is a compact operator^{} iff there is a sequence $\{{F}_{n}\}$ of finite rank operators with $\parallel T-{F}_{n}\parallel \to 0$.

###### Proof.

$(\Rightarrow )$: Assume $T$ is compact^{} on $\mathscr{H}$ and $\{{e}_{n}\}$ is an orthonormal basis of $\mathscr{H}$. Define:

${P}_{n}f$ | $={\displaystyle \sum _{k=0}^{n}}\u27e8f,{e}_{k}\u27e9{e}_{k}$ |

It is clear that the ${P}_{n}$ have finite rank and that we have $\parallel {P}_{n}f\parallel \le \parallel f\parallel $ for all $n\in \mathbb{N}$, $f\in \mathscr{H}$.

Let $\mathcal{B}$ be the unit ball in $\mathscr{H}$. We have that ${P}_{n}\to I$ pointwise. Since the ${P}_{n}$ are contractive they are equicontinuous, hence ${P}_{n}$ converges uniformly to $I$ on compact sets, and in particular on $\overline{T(\mathcal{B})}$, which is compact by assumption^{}.
Therefore ${P}_{n}T\to T$ uniformly on $\mathcal{B}$, hence $\parallel {P}_{n}T-T\parallel \to 0$.
Since ${P}_{n}T$ is bounded^{} and of finite rank the first direction follows.

$(\Leftarrow )$: Now let $\{{F}_{n}\}$ be a sequence of bounded operators^{} of finite rank with $\parallel T-{F}_{n}\parallel \to 0$.
We have to show that $T(\mathcal{B})$ is relatively compact in $\mathscr{H}$. This is equivalent^{} to $T(\mathcal{B})$ being totally bounded^{} in $\mathscr{H}$.
So we are left to show that for all $\u03f5>0$ there is an $\u03f5$-net ${x}_{1},\mathrm{\cdots},{x}_{n}\in \mathscr{H}$ so that:

$T(\mathcal{B})$ | $\subseteq {\displaystyle \bigcup _{k=1}^{n}}{B}_{\u03f5}({x}_{k})$ |

So choose $\u03f5>0$ and $n\in \mathbb{N}$ fixed so that:

$$ |

Choose ${x}_{1},\mathrm{\cdots},{x}_{m}\in \mathscr{H}$ with:

${F}_{n}(\mathcal{B})$ | $\subseteq {\displaystyle \bigcup _{k=1}^{m}}{B}_{\frac{\u03f5}{2}}({x}_{k})$ |

Hence (by the triangle inequality):

$T(\mathcal{B})$ | $\subseteq {\displaystyle \bigcup _{k=1}^{m}}{B}_{\u03f5}({x}_{k})$ |

and we are done. ∎

Title | finite rank approximation on separable Hilbert spaces |
---|---|

Canonical name | FiniteRankApproximationOnSeparableHilbertSpaces |

Date of creation | 2013-03-22 18:23:18 |

Last modified on | 2013-03-22 18:23:18 |

Owner | karstenb (16623) |

Last modified by | karstenb (16623) |

Numerical id | 10 |

Author | karstenb (16623) |

Entry type | Theorem |

Classification | msc 46B99 |