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 ( D , C ) {\displaystyle H^{2}(\mathbb {D} ,\mathbb {C} )} . It states that each such space is of the form
θ H 2 ( D , C ) , {\displaystyle \theta H^{2}(\mathbb {D} ,\mathbb {C} ),}
for some inner function θ {\displaystyle \theta } . The result is a significant connection between operator theory with complex analysis.
See also Hardy space H2 Unilateral shift operator
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 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.