THE SPACE H~2 OF ANALYTIC FUNCTION VECTORS
Author(s): Ni Zhong-ren
Pages: 498-504
Year: 1985 Issue: 4
Journal: Acta Mathematicae Applicatae Sinica
Keyword: 解析函数; H~2; 最大公因子; 内函数; 函数矩阵; 闭子空间; 单位圆; 不变子空间; 内因子; 特征矩阵;
Abstract: Beurling described in 1949 all the invariant subspaces for the operator"multiplication by z"on the Hilbert space H~2 of analytic ftnctions in the unit disc.Let H~2 be the space composed of analytic functions of class H~2 in the unit disc,S be a non-trivial closed subspace of H~2, and z be a complex wariable.Beurling obtained the following conclusions:(1) zS(?)S if and only if S has the form S=F(z)H~2,where F(z)is an inner function;(2) when S=F(z)H~2,F(z) is the greatest common divisor of the inner parts of the functions in S;(3) let h(z) be a function in H~2.Then z~nh(z),n=0,1,2,…span H~2 if and only if h(z) is an outer function.In this paper,Beurling’s conclusions have been expanded to the spaces of analytic function vectors.
Pages: 498-504
Year: 1985 Issue: 4
Journal: Acta Mathematicae Applicatae Sinica
Keyword: 解析函数; H~2; 最大公因子; 内函数; 函数矩阵; 闭子空间; 单位圆; 不变子空间; 内因子; 特征矩阵;
Abstract: Beurling described in 1949 all the invariant subspaces for the operator"multiplication by z"on the Hilbert space H~2 of analytic ftnctions in the unit disc.Let H~2 be the space composed of analytic functions of class H~2 in the unit disc,S be a non-trivial closed subspace of H~2, and z be a complex wariable.Beurling obtained the following conclusions:(1) zS(?)S if and only if S has the form S=F(z)H~2,where F(z)is an inner function;(2) when S=F(z)H~2,F(z) is the greatest common divisor of the inner parts of the functions in S;(3) let h(z) be a function in H~2.Then z~nh(z),n=0,1,2,…span H~2 if and only if h(z) is an outer function.In this paper,Beurling’s conclusions have been expanded to the spaces of analytic function vectors.
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