Beurling–Lax theorem

Short description: Theorem in mathematics

In mathematics, the Beurling–Lax theorem is a theorem due to (Beurling 1948) and (Lax 1959) which characterizes the shift-invariant subspaces of the Hardy space $H^2(\mathbb{𝔻},\mathbb{ℂ})$. It states that each such space is of the form

$\theta H^2(\mathbb{𝔻},\mathbb{ℂ}),$

for some inner function $\theta$.

The result is a significant connection between operator theory with complex analysis.

See also

• Hardy space
• H²
• Unilateral shift operator

References

• Hazewinkel, Michiel, ed. (2001), "Beurling-Lax theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Beurling, A. (1948), "On two problems concerning linear transformations in Hilbert space", Acta Math. 81: 239–255, doi:10.1007/BF02395019 
• Lax, P.D. (1959), "Translation invariant spaces", Acta Math. 101 (3–4): 163–178, doi:10.1007/BF02559553 
• 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.