Invariant subspaces of the operator of multiple differentiation
type: Item
instance of: scholarly article
title: Invariant subspaces of the operator of multiple differentiation
zbMATH Open document ID: 0541.47006
DOI: 10.1007/BF01158283
author: S. G. Merzlyakov
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: 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 \(p_ n\equiv 1\).
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 3859799
zbMATH Keywords: operator of multiple differentiation
zbMATH Keywords: root vectors
zbMATH Keywords: space of entire functions
zbMATH Keywords: spectral synthesis
zbMATH Keywords: invariant subspace
MaRDI profile type: Publication
cites work: Théorie générale des fonctions moyenne-périodiques
cites work: Q5531375
cites work: Q5332414
cites work: Q5583334
description: scientific article; zbMATH DE number 3859799