View source for Beurling–Lax theorem

In [[mathematics]], the ''' Beurling–Lax theorem''' is a [[theorem]] due to {{harvtxt|Beurling|1948}} and {{harvtxt|Lax|1959}} which characterizes the [[shift operator|shift]]-[[invariant subspace|invariant]] subspaces of the [[Hardy space]] <math>H^2(\mathbb{D},\mathbb{C})</math>. It states that each such space is of the form

: <math> \theta H^2(\mathbb{D},\mathbb{C}), </math>

for some [[inner function]] <math>\theta</math>.

==See also==
*[[H square|H<sup>2</sup>]]

==References==
*{{eom|title=Beurling-Lax theorem|last=Ball|first=J. A. }}
*{{citation|authorlink=Arne Beurling|first=A. |last=Beurling|title=   On two problems concerning linear transformations in Hilbert space|journal= Acta Math. |volume= 81  |year=1948|pages= 239–255	|doi=10.1007/BF02395019|mr=0027954|doi-access=free}}
*{{citation|authorlink=Peter Lax|first=P.D.|last= Lax|title=Translation invariant spaces|journal=  Acta Math. |volume=101|issue=3–4  |year=1959|pages= 163–178|doi=10.1007/BF02559553|mr=0105620|doi-access=free}}
* Jonathan R. Partington, ''Linear Operators and Linear Systems, An Analytical Approach to Control Theory'', (2004) London Mathematical Society Student Texts '''60''', Cambridge University Press.
* Marvin Rosenblum and James Rovnyak, ''Hardy Classes and Operator Theory'', (1985) Oxford University Press.

{{DEFAULTSORT:Beurling-Lax theorem}}
[[Category:Hardy spaces]]
[[Category:Theorems in analysis]]
[[Category:Invariant subspaces]]


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