SPECTRUM PROPERTIES OF THE KOOPMAN AND FROBENIU-PERRON OPERATES

Consider the measure space (X, A, μ). Let S: X → X be a nonsingular transformation on this space and assumes that is the Frobenius Perron operator associated with s. Also we will assume that Us is Koopman operator corresponding to s. Properties of these two operators, namely Ps and Us have been considered by many authors. For more details see [5], [7], [8] and [9]. In this paper we will present some special aspects concerning the spectrum of Ps. Also is shown that the spectrum of is a cyclic subset of the unit disk D. In connection with the Koopman operator Us it is shown that under certain conditions, if μ is a regular measure, then Us from L∞ (X) to itself is an isometric.