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Integral Equations and Operator Theory

, Volume 63, Issue 2, pp 181–215 | Cite as

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

  • I. Yu. Domanov
  • M. M. Malamud
Article

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.

Keywords.

Riemann-Liouville operator invariant subspace hyperinvariant subspace commutant double commutant 

Mathematics Subject Classification (2000).

Primary 47A15, 47A16, 47L80 Secondary 47L10 

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Copyright information

© Birkhaueser 2009

Authors and Affiliations

  1. 1.Mathematical Institute, AS CRPraha 1Czech Republic
  2. 2.Institute of Applied Mathematics and MechanicsDonetskUkraine

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