Beurling–Lax theorem

In mathematics, the Beurling–Lax theorem is a theorem due to Beurling (1949) and Lax (1959) which characterizes the shift-invariant subspaces of the Hardy space ${\displaystyle H^{2}(\mathbb {D} ,\mathbb {C} )}$. It states that each such space is of the form

${\displaystyle \theta H^{2}(\mathbb {D} ,\mathbb {C} ),}$

for some inner function ${\displaystyle \theta }$.

See also

• H²

References

• Ball, J. A. (2001) [1994], "Beurling-Lax theorem", Encyclopedia of Mathematics, EMS Press
• Beurling, A. (1949), "On two problems concerning linear transformations in Hilbert space", Acta Math., 81: 239–255, doi:10.1007/BF02395019, MR 0027954
• Lax, P.D. (1959), "Translation invariant spaces", Acta Math., 101 (3–4): 163–178, doi:10.1007/BF02559553, MR 0105620
• 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.