AN EXAMPLE OF A MINIMAL ACTION OF THE FREE SEMI-GROUP F-2(+) ON THE HILBERT SPACE

作者：Grivaux Sophie^*; Roginskaya Maria

来源：Mathematical Research Letters, 2013, 20(4): 695-704.

DOI：10.4310/MRL.2013.v20.n4.a7

• Invariant Subspace and Invariant Subset Problems on Hilbert spaces
• hypercyclic vectors
• orbits of linear operators
• Read%26apos
• s type operators
• minimal action of groups on a Hilbert space

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator T on a separable infinite-dimensional Hilbert space H such that the orbit {T(n)x; n %26gt;= 0} of every non-zero vector x is an element of H under the action of T is dense in H. We show that there exists a bounded linear operator T on a complex separable infinite-dimensional Hilbert space H and a unitary operator V on H, such that the following property holds true: for every non-zero vector x is an element of H, either x or V-x has a dense orbit under the action of T. As a consequence, we obtain in particular that there exists a minimal action of the free semi-group with two generators F-2(+) on a complex separable infinite-dimensional Hilbert space H. The proof involves Read%26apos;s type operators on the Hilbert space, and we show in particular that these operators - which were potential counterexamples to the Invariant Subspace Problem on the Hilbert space - do have non-trivial invariant closed subspaces.

• 出版日期2013