Invariant and hyperinvariant subspaces of the operator \(J^\alpha\) in the Sobolev spaces \(W^s_p[0,1]\). (Q1889522)

The author describes the lattices \(\operatorname{Lat} J_{s}^{\alpha}\) and \(\operatorname{Hyplat} J_{s}^{\alpha}\) of invariant and hyperinvariant subspaces of the operator \(J_{s}^{\alpha}\) acting on the Sobolev space \(W_{p}^{s}[0,1]\) \((s >0)\) and studies the operator algebra \(\operatorname{Alg} J_{s}^{\alpha}\), the commutant \(\{J_{s}^{\alpha}\}'\), and the bicommutant \(\{ J_{s}^{\alpha} \}''\). For this, he develops the methods of \textit{I. Yu. Domanov} and \textit{M. M. Malamud} [Linear Algebra Appl. 348, 209--230 (2002; Zbl 1006.47003)].