Beurling theorem

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