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On the spectral analysis of direct sums of Riemann-Liouville operators in Sobolev spaces of vector functions. (English) Zbl 1220.47009
Summary: Let \(J^{\alpha }_{k }\) 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 \(\{A _{k }\}^{\prime}\), the double commutant \(\{A_k\}^{\prime\prime}\) and the algebra \({\mathtt Alg}\, A _{k }\). Moreover, we describe the lattices \({\mathtt Lat}\, A _{k }\) and \({\mathtt HypLat}\, A _{k }\) 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 \({\mathtt Cyc}\, A _{k }\) of its cyclic subspaces. In passing, we present a simple counterexample for the implication \[ {\mathtt HypLat}\,(A \oplus B) = {\mathtt HypLat}\, A \oplus {\mathtt HypLat}\, B \Rightarrow {\mathtt Lat}\,(A \oplus B) = {\mathtt Lat}\,A \oplus {\mathtt Lat}\,B \] to be valid.

MSC:
47A15 Invariant subspaces of linear operators
47A16 Cyclic vectors, hypercyclic and chaotic operators
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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