Invariant subspaces of the operator of multiple differentiation
Let H be the space of entire functions of one complex variable, with the topology of uniform convergence on compact sets, and let A be a continuous linear operator on H. Further, let W be a (closed) subspace of H invariant for A, i.e., \(AW\subset W.\) We say that W allows a spectral synthesis of A if W coincides with the closed linear span of the root vectors of \(A| W\). \textit{L. Schwartz} has proved [Ann. Math., II. Ser. 48, 857-929 (1947; Zbl 0030.15004)] that if D denotes the operator of differentiation on H then every closed D-invariant subspace of H allows a spectral synthesis of D. In the present paper it is proved that the same assertion holds for any operator of the form \(A=\sum^{n}_{k=0}p_ kD^ k,\) \(n\geq 1\), where \(p_ k\in H\) and \(p_ n\equiv 1\).
scientific article; zbMATH DE number 3859799