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

Abstract.

Let J ^α _k be a real power of the integration operator J _k defined on the Sobolev space W ^k _p [0, 1]. We investigate the spectral properties of the operator $A_{k} = \bigoplus^{n}_{j=1} \lambda_{j}J^{\alpha}_{k}$ defined on $\bigoplus^{n}_{j=1}W^{k}_{p} [0, 1]$. Namely, we describe the commutant {A _k }′, the double commutant $\{A_k\}\prime\prime$ 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

$${\tt HypLat}(A \oplus B) = {\tt HypLat}\, A \oplus {\tt HypLat}\, B \Rightarrow {\tt Lat}(A \oplus B) = {\tt Lat}\,A \oplus {\tt Lat}\,B$$

to be valid.