Beurling–Lax theorem

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 $\theta$.

See also

• H²

References

• Ball, J. A. (2001) [1994], "Beurling-Lax theorem", Encyclopedia of Mathematics, EMS Press
• Beurling, A. (1948), "On two problems concerning linear transformations in Hilbert space", Acta Math., 81: 239–255, doi:10.1007/BF02395019, MR 0027954239-255&rft.date=1948&rft_id=info:doi/10.1007/BF02395019&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0027954#id-name=MR&rft.aulast=Beurling&rft.aufirst=A.&rfr_id=info:sid/en.wikipedia.org:Beurling–Lax theorem" class="Z3988">
• Lax, P.D. (1959), "Translation invariant spaces", Acta Math., 101 (3–4): 163–178, doi:10.1007/BF02559553, MR 01056203–4&rft.pages=163-178&rft.date=1959&rft_id=info:doi/10.1007/BF02559553&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0105620#id-name=MR&rft.aulast=Lax&rft.aufirst=P.D.&rfr_id=info:sid/en.wikipedia.org:Beurling–Lax theorem" class="Z3988">
• 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.