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A Survey of the Dynamics of Cohyponormal Operators

For a linear operator T on a Hilbert space H and a vector x in H, the orbit of x (under T) is the sequence ${T^nx}$.  The dynamics of a linear operator T involves the behavior of the orbits of T.  While subnormal operators and hyponormal operators have rather tame orbits - their orbits always converge to infinity in norm or to 0 weakly - the orbits of a cosubnormal operator or a cohyponormal operator can actually be dense in the whole Hilbert space H.  Recall that an operator is cosubnormal or cohyponormal if its adjoint is subnormal or hyponormal, respectively.  In fact the first example of a linear operator on a Hilbert space having a dense orbit (due to Rolewicz) is twice the Backward shift - which is indeed cosubnormal.

In this paper we survey some results by the author and co-authors regarding which cohyponormal operators have a "small" set with dense orbit.  Surprisingly, we can characterize in terms of spectral properties when a cohyponormal operator has a "small" set with dense orbit.  By a "small" set we mean any of the following:  a point, a 1-dimensional subspace, a finite dimensional subspace, a bounded set, or a bounded countable separated set.

The case of having a finite dimensional subspace with dense orbit is not complete.  There are necessary conditions and sufficient conditions, but they are not quite the same.  The case of a 2-dimensional subspace is completely understood.

The paper concludes by discussing some "complicated" orbits for linear operators.  For example, constructing linear operators with orbits dense in Cantor sets, Julia sets, and other geometrically complicated sets.  Constructing orbits that are d-dense but not dense.  We say that an orbit is d-dense if the orbit comes within a distance d of every point in the space.  For every d > 0, there are d-dense orbits that are not dense.  However, if an operator has a d-dense orbit, then it must be hypercyclic.  We also discuss weakly dense orbits and the fact that there exists orbits that are weakly dense but not norm dense.