Invariant subspaces of the shift operator in weighted Hilbert space

B. I. Korenblum

Sbornik. Mathematics, Tome 18 (1972) no. 1, pp. 111-138

Résumé

A complete description is given of the closed ideals of the algebra $H_1^2$ of functions $\widehat x(z)$ which are regular in the circle $U$ ($|z|1$) and such that $\widehat x'\in H^2$, with the norm $$ \|\widehat x\|_{H_1^2}=(\|\widehat x\|_{H^2}^2+\|\widehat x'\|_{H^2}^2)^{1/2} $$ and the usual multiplication. This is equivalent to a description of the invariant subspaces of the one-sided shift operator on the weighted Hilbert space of sequences with weights $p_k=1+k^2$ ($k=0,1,\dots$). It is shown that each closed ideal $I$ of the algebra $H_1^2$ has the form $I=\overline I\cap A$, where $\overline I$ is the closure of $I$ in the space $A$ of functions which are regular in $U$ and continuous in $\overline U$ with the uniform norm. Thus the ideals of the algebra $H_1^2$ have a structure similar to the structure of the ideals of the algebra $A$: each ideal $I$ is uniquely determined by an interior function $G$, which is the greatest common divisor of the interior parts of the functions $\widehat x\in I$, and the set $K\subset\partial U$ of the common zeros of the functions $\widehat x\in I$. Bibliography: 19 titles.