Invariant and hyperinvariant subspaces of some Volterra operators in Sobolev spaces and related operator algebras
by
Ignat Domanov
Donetsk State University, Ukraine
Coauthors: M.M. Malamud (Donetsk State University)

In our investigation we describe the lattices Lat A and Hyplat A and the set Cyc A of cyclic subspaces for an operator A=J^\alpha\otimesB being a tensor product of J^\alpha defined on X and an arbitrary n×n nonsingular diagonal matrix B=diag(\lambda₁, ..., \lambda_n ). Here X=W_p^k[0, 1],  p in [1, \infty), k in Z₊\{0}.

Descriptions of the operator algebras {A}',  {A}'' and AlgA, are presented too. Here as usual {A}' and {A}'' are the commutant and the double commutant of A respectively, and AlgA stands for the weakly closed subalgebra of L(X\otimesCⁿ) generated by A and the identity I.

These results on operator algebras allow us to describe the lattice HyplatA of hyperinvariant subspaces as well as to construct counterexamples to some hypotheses which are valid for C₀-contractions.

Most of the results are new even in the scalar case.

1) The operator J^\alpha is unicellular on W₂^k[0, 1] (with k >= 2) if and only if \alpha = 1.

2) Each R in {J^\alpha}' is of the form : R=I+R₁, where R₁ in \cap _{q > 1}S_q.

3) The algebra Alg J^m is a proper subalgebra of the double commutant {J^m}'' and the dimension d_m of the quotient algebra {J^m}''/ Alg J^m is : d_m=k-1-[[(k-1)/m]] for m in Z₊\{0}.

4) Hyplat J_k^\alpha is unicellular (i.e. linearly ordered) as distinct from Lat J_k^\alpha (for k >= 2);

We generalize also some results on {A}' and {A}'' from [1].

References:

1. Malamud M.M. Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators -Operator Theory : Advances and Applicanions, Vol. 102, p. 143-167

(T)

Date received: December 21, 1999