Hypercyclic tuples of operators and somewhere dense orbits

Abstract

In this paper we prove that there are hypercyclic (n + 1)-tuples of diagonal matrices on C n and that there are no hypercyclic n-tuples of diagonalizable matrices on C n . We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.

Dynamics of tuples of matrices

“…Let us prove assertion (i). The case of A 1 , A 2 both diagonal is covered by Feldman; see [4]. Assume that A 1 is diagonal and A 2 is in Jordan form; i.e.…”

“…In this section we start with the following special case of Corollary 3.5 in [4], due to Feldman, which will be of use to us in the following. …”

“…For an account of results, comments and an extensive bibliography on hypercyclicity we refer to [1], [5], [6] and [7]. For results concerning the dynamics of tuples of operators see [2], [3], [4] and [9].…”

“…In [4] Feldman showed, among other things, that in C n there exist diagonalizable (n + 1)-tuples of matrices having dense orbits. In addition he proved that there is no n-tuple of diagonalizable matrices on R n or C n that has a somewhere dense orbit.…”

Strong mixing measures and invariant sets in linear dynamics

“…A type of linear universality which has attracted the attention in recent years is the dynamics of tuples of operators introduced by Feldman [55]. More precisely, given a commuting tuple (T 1 , .…”