On reproducing subspaces of Volterra operators
type: Item
instance of: scholarly article
title: On reproducing subspaces of Volterra operators
zbMATH Open document ID: 0894.47038
published in: Doklady Mathematics
publication date: 1998-09-08T00:00:00Z
review text: The theory of Volterra integral operators in \(L_2[0, 1]\) is well developed, much less is known about these operators in the spaces of vector-valued functions \(L_2[0, 1]\otimes \mathbb{C}^n\). This paper deals with the study of tensor products \(K\otimes B\) of operators \(K\) and \(B\) in \(L_p[0, 1]\otimes\mathbb{C}^n\), where the first operator is a Volterra operator in \(L_p[0, 1]\), and the second operator is diagonalizable in \(\mathbb{C}^n\). In some cases, we describe their invariant and reproducing subspaces, obtain criteria of cyclicity, and determine the multiplicity of the spectrum. The obtained results are new for the operator \(J\oplus J\) as well. We adduce some results on the similarity to the operators \(J^\alpha\otimes B\) and the stability of the properties of cyclicity and unicellularity under nilpotent perturbations, some of which are new even for \(n= 1\) (see Theorems 7 and 8). It should be noted that Theorem 7 contains the answer to the question of \textit{I. Ts. Gokhberg} and \textit{M. G. Krein} [``Theory of Volterra operators in Hilbert space and its applications'' (in Russian), Moskva (1967; Zbl 0168.12002), p. 421].
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A65
zbMATH DE Number: 1155306
zbMATH Keywords: Volterra integral operators
zbMATH Keywords: spaces of vector-valued functions
zbMATH Keywords: tensor products
zbMATH Keywords: invariant and reproducing subspaces
zbMATH Keywords: cyclicity
zbMATH Keywords: multiplicity of the spectrum
zbMATH Keywords: unicellularity
zbMATH Keywords: nilpotent perturbations
author: M. M. Malamud
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1155306