Beurling–Lax theorem

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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 . It states that each such space is of the form

for some inner function .

See also[edit]


  • Ball, J. A. (2001) [1994], "b/b120200", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • 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.