On reproducing subspaces of Volterra operators
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].
scientific article; zbMATH DE number 1155306
scientific article