zbMATH — the first resource for mathematics

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.
MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators 47A15 Invariant subspaces of linear operators 47G10 Integral operators 47B38 Linear operators on function spaces (general)
Full Text: