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 the Sobolev space W-p(k) [0, 1]. We investigate the spectral properties of the operator A(k) = circle plus(n)(j=1) lambda(j)J(k)(alpha) k defined on circle plus(n)(j=1) W-p(k)[0, 1]. Namely, we describe the commutant {Ak}', the double commutant {Ak}'' 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 circle plus B) - HypLat A circle plus HypLat B double right arrow Lat(A circle plus B) = Lat A circle plus Lat B to be valid.