Approximate spectral synthesis in the Bergman space
Recall that a space of analytic functions on the unit disc is called \(z\)-invariant if it is invariant under the operator \(M\) of multiplication by the independent variable. For the Bergman space \(L^2_a\) it is proved that any \(z\)-invariant subspace of index \(1\) admits strong approximate spectral cosynthesis, i.e.\ it may be written as a lower limit of a sequence of finite-codimensional \(z\)-invariant subspaces such that also the annihilator is the lower limit of the corresponding annihilators. Moreover, any \(z^*\)-invariant subspace of the dual (i.e.\ invariant under \(M^*\)) admits weak approximate spectral synthesis, i.e.\ it may be written as a lower limit of a sequence of finite-dimensional \(z^*\)-invariant subspaces. This proves a conjecture of \textit{N. K. Nikol'skij} [J. Sov. Math. 26, 2185-2186 (1984; Zbl)]. The result has some applications to approximation by rational functions.
scientific article; zbMATH DE number 1425271
scientific article