Sequences of vectors that are orbits of operators (Q2488766)

Let \(T\) be a linear operator on a vector space \(X\) and \(x\) a fixed element in \(X\). An orbit of \(T\) will be the sequence in \(X\) generated by all the powers of \(T\) applied to \(x\). This paper is concerned with the question when a sequence of vectors from \(X\) is the orbit of some linear operator. For this, in the first section some algebraic results are presented. In the second section, the main theorem (Theorem 2.3) is proved. Equivalent conditions for a sequence in a Hilbert space to be the orbit of a unitary operator are obtained. The main theorem gives a nice look on stochastic processes in terms of their orbit characterization.