Atlas: The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on $\bf Z$ by Aharon Atzmon

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18th International Conference on Operator Theory
June 27 - July 1, 2000
University of the West
Timisoara, Romania

Organizers
Dumitru Gaspar, Traian Ceausu, Aurelian Craciunescu, Aurelian Gheondea, Radu-Nicolae Gologan, Ciprian Pop, Dan Popovici, Nicolae Suciu, Alexandru Terescenco, Dan Timotin, Flavius Turcu

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The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on Z
by
Aharon Atzmon
Tel Aviv University

If X is a translation invariant Banach space of complex sequences on the integer group Z, then from the point of view of harmonic analysis, it is natural to study the (closed) translation invariant subspaces of X. The first problem that arises in this context, is whether X has any nontrivial translation invariant subspace. This problem is open even for weighted l^p spaces on Z, and for this case it is equivalent to the hyperinvariant subspace problem for bilateral weighted shifts on l^p. We shall prove that the solution is positive for even weights and 1 < p < \infty. We deduce this from the following more general result.

Theorem 1. Assume that X is a reflexive translation invariant Banach space of complex sequences on Z, that contains all finitely supported sequences, in which the coordinate functionals are continuous, and for every sequence {c(n)} in the space, the sequences {[`c(n)]} and {c(-n)} are also in the space. Then X has a nontrivial translation invariant subspace.

The proof of the theorem is based on the following intermediate result, which was inspired by a paper of Simonic [2].

Theorem 2. Let E be a reflexive real Banach space of dimension greater than one, and assume that A is an operator on E, such that for some non-zero vectors u in E and v in E^*, { <Aⁿ u, v>}_n=0^\infty is a moment sequence of a finite positive Bored measure on a bounded interval on the real line. Then A has a nontrivial invariant subspace.

Using this result, we also show that every weighted bi-shift on l² (Z₊), as defined in [1], has a nontrivial invariant subspace. This provides an affirmative answer to the main problem raised in that paper.

References

[1] A. Atzmon and M. Sodin, Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type, J. Funct. Anal. 169 (1999), 164-188.

[2] A. Simonic, An extension of Lomonosov's techniques to non-compact operators, Trans. Amer. Math. Soc. 348 (1996), 975-995.

Date received: May 4, 2000