On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

Let Jαk be a real power of the integration operator Jk defined on the Sobolev space Wkp[0, 1]. We investigate the spectral properties of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{k} = \bigoplus^{n}_{j=1} \lambda_{j}J^{\alpha}_{k}$$\end{document} defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigoplus^{n}_{j=1}W^{k}_{p} [0, 1]$$\end{document}. Namely, we describe the commutant {Ak}′, the double commutant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A_k\}\prime\prime$$\end{document} and the algebra Alg Ak. Moreover, we describe the lattices Lat Ak and HypLat Ak of invariant and hyperinvariant subspaces of Ak, respectively. We also calculate the spectral multiplicity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{A_k}$$\end{document} of Ak and describe the set Cyc Ak of its cyclic subspaces. In passing, we present a simple counterexample for the implication \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tt HypLat}(A \oplus B) = {\tt HypLat}\, A \oplus {\tt HypLat}\, B \Rightarrow {\tt Lat}(A \oplus B) = {\tt Lat}\,A \oplus {\tt Lat}\,B$$\end{document} to be valid.