TWO-SIDED AND ONE-SIDED INVERTIBILITY OF WIENER-TYPE FUNCTIONAL OPERATORS WITH A SHIFT AND SLOWLY OSCILLATING DATA

Let a be an orientation-preserving homeomorphism of [0, infinity] onto itself with only two fixed points at 0 and infinity, whose restriction to R+ = (0, infinity) is a diffeomorphism, and let U-alpha be the isometric shift operator acting on the Lebesgue space L-P(R+) with p is an element of [1, infinity] by the rule U(alpha)f = (alpha')(1/P)(f o alpha). We establish criteria of the two-sided and one-sided invertibility of functional operators of the form A = Sigma (k is an element of Z) akU(alpha)(k) where parallel to A parallel to w = Sigma (k is an element of Z) parallel to a(k)parallel to L-infinity(R+) <infinity, on the spaces L-P(R+) under the assumptions that the functions log(/' and ak for all k is an element of Z are bounded and continuous on R+ and may have slowly oscillating discontinuities at 0 and infinity. The unital Banach algebra U-w of such operators is inverse-closed: if A is an element of U-w is invertible on L-P(R+) for p is an element of [1, infinity], then A(-1) is an element of U-w. Obtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called discrete operators on the spaces l(P) and in terms of conditions related to the fixed points of a and the orbits {alpha(n)(t) : n is an element of Z} of points t is an element of R+