# Beurling theorem

From Encyclopedia of Mathematics

Let $f$ be a function in the Hardy class $H^2$ (cf. also Hardy classes). The vector space spanned by the functions $e^{in\theta}f$, $n\geq0$, is dense in $H^2$ if and only if $f$ is an outer function (cf. also Hardy classes).

This follows from the characterization of closed shift-invariant subspaces in $H^2$ as being of the form $gH^2$ with $g$ an inner function.

See Beurling–Lax theorem for further developments.

#### References

[a1] | B. Sz.-Nagy, C. Foias, "Harmonic analysis of operators on Hilbert spaces" , North-Holland (1970) pp. 104 |

[a2] | B. Beauzamy, "Introduction to operator theory and invariant subspaces" , North-Holland (1988) pp. 194 |

[a3] | W. Mlak, "Hilbert spaces and operator theory" , Kluwer Acad. Publ. (1991) pp. 188; 190 |

**How to Cite This Entry:**

Beurling theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Beurling_theorem&oldid=33509

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article