Chaos for weighted shift operators and invariant subspace problem

Abstract

One of the most important open problems in operator theory on Hilbert space is the invariant subspace problem. The invariant subspace problem is the easy question: Does every bounded operator T on a separable Hilbert space over have a non-trivial invariant subspace? The problem is solved for finite dimensional complex vector spaces and on a non-separable infinite-dimensional complex Hilbert space. The invariant subspace problem is solved In the Banach space by Per Enflo in 1976. The relation between the invariant subspace and the topologically transitivity is that if the dynamical system is irreduciple (cannot be decayed into two disjoint invariant subsets) then it is topologically transitive. intuitively, a topologically transitive map has points which finally move under iteration from arbitrarly small neighbourhood to any other.

A cyclic vector for a bounded operator on a Banach space is one that its orbit under that operator has dense linear span. If the orbit itself is dense, the vector is called hyper-cyclic. A vector is super-cyclic for an operator if the scalar- multiples of the elements in its orbit are dense. Thus hyper-cyclic implies super-cyclic, which in turn implies cyclic. The weight of cyclic vectors derives from the learning of invariant subspaces. The closed linear span of the orbit of a vector is the least closed subspace, invariant under the operator, that contains the vector. Thus an operator has no non-trivial invariant subspaces if and only if each non-zero vector is cyclic. Similarly, an operator has no non-trivial closed invariant subset if and only if each non-zero vector is hyper-cyclic. The first examples of hypercyclic operators are due to Birkhoff, MacLane and Rolewicz. In 1929 Birkhoff [34] showed the hyper-cyclicity of the translation operators