On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions
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چکیده
Let Jα k be a real power of the integration operator Jk defined on Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator Ak = ⊕n j=1 λjJ α k defined on ⊕n j=1W k p [0, 1]. Namely, we describe the commutant {Ak} ′, the double commutant {Ak} ′′ and the algebra AlgAk. Moreover, we describe the lattices LatAk and HypLatAk of invariant and hyperinvariant subspaces of Ak, respectively. We also calculate the spectral multiplicity μAk of Ak and describe the set CycAk of its cyclic subspaces. In passing, we present a simple counterexample for the implication HypLat(A⊕B) = HypLatA⊕ HypLatB ⇒ Lat(A⊕B) = LatA⊕ LatB to be valid.
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