Invariant and hyperinvariant subspaces of an operator Jα and related operator algebras in Sobolev spaces Original Research Article

We investigate the spectral properties of the complex powers Jkα of the integration operator Jk defined on a Sobolev space Wpk[0,1]. Namely, we describe the lattice Lat Jkα of invariant subspaces and the lattice Hyplat Jkα of hyperinvariant subspaces of the operator Jαk. In particular, it turns out that the operator Jkα is unicellular on Wpk[0,1] iff either k=1 or α=1 though the lattice Hyplat Jkα is unicellular (i.e. linearly ordered) for each k and α>0. The operator algebra Alg Jkα, commutant {Jkα}′ and double commutant {Jkα}″ are investigated too. In particular we prove that the operator Akα=Jkαcircled plusJ(0;k)α satisfies the Neumann–Sarason identity Alg Akα={Akα}″ though Jkα does not. Thus we present a simple counterexample for the implication image Alg(Acircled plusB)={Acircled plusB}″, Alg B={B}″ implies Alg A={A}″to be valid.