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

Let $J_k^{\alpha }$ be a real power of the integration operator $J_k$ defined on Sobolev space $W_p^k[0,1]$. We investigate the spectral properties of the operator $A_k={⨁}_{j=1}^n{\lambda }_j{J}_k^{\alpha }$ defined on ${⨁}_{j=1}^n{W}_p^k[0,1]$. Namely, we describe the commutant $\{A_k{\}}^{\mathrm{\prime }}$, the double commutant $\{A_k{\}}^{\mathrm{\prime }\mathrm{\prime }}$ and the algebra $\text{Alg}{A}_k$. Moreover, we describe the lattices $\text{Lat}{A}_k$ and $\text{Hyplat}{A}_k$ of invariant and hyperinvariant subspaces of $A_k$, respectively. We also calculate the spectral multiplicity ${\mu }_{A_k}$ of $A_k$ and describe the set $\text{Cyc}{A}_k$ of its cyclic subspaces. In passing, we present a simple counterexample for the implication \Hyplat(A\oplus B)=\Hyplat A\oplus \Hyplat B\Rightarrow \Lat(A\oplus B)=\Lat A\oplus \Lat B to be valid.