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 $A_k=\bigoplus_{j=1}^n \lambda_j J_k^\alpha$ defined on $\bigoplus_{j=1}^n W_p^k[0,1]$. Namely, we describe the commutant $\{A_k\}'$, the double commutant $\{A_k\}''$ and the algebra $\Alg A_k$. Moreover, we describe the lattices $\Lat A_k$ and $\Hyplat A_k$ 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 $\Cyc A_k$ 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.