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On the spectral multiplicity of some Volterra operators in Sobolev spaces. (English. Russian original) Zbl 1158.47301
Math. Notes 72, No. 2, 275-280 (2002); translation from Mat. Zametki 72, No. 2, 306-311 (2002).
Define the operator \[ J^{\alpha}:f(x)\mapsto\int_{0}^x f(t){\frac{(x-t)^{\alpha-1}}{\Gamma (\alpha)}}\,dt, \] where the real part of \(\alpha\) is positive. For a diagonal \(n\times n\) matrix \(B\), the operator \(J^{\alpha}\otimes B\) is considered on a direct sum of Sobolev spaces \(\bigoplus_{i=1}^{ l} W_{p}^{k_i}[0,1]\). Cyclic invariant subspaces of \(J^{\alpha}\otimes B\) are described, under mild conditions on the diagonal entries of \(B\). In particular, a criterion is given, in terms of ranks of certain matrices, for a finite set of elements of \(\bigoplus_{i=1}^{l} W_{p}^{k_i}[0,1] \) to generate a cyclic \(J^{\alpha}\otimes B\)-invariant subspace. No proofs are given.
47A16 Cyclic vectors, hypercyclic and chaotic operators
47A15 Invariant subspaces of linear operators
47G10 Integral operators
47B38 Linear operators on function spaces (general)
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