ILLINOIS JOURNAL OF MATHEMATICS Volume 40, Number 1, Spring 1996 SPECTRAL PROPERTIES OF WEIGHTED COMPOSITION OPERATORS AND HYPERBOLICITY OF LINEAR SKEW-PRODUCT FLOWS YURI LATUSH KIN 1. Introduction A weighted composition operator is an operator T that acts by the rule (Tf)(x) a(x)f(cpx) on a space of vector-valued functions f, defined on a set X. Here p is a given mapping of X, and a(.) is a given operator-valued function. These operators have been studied with different purposes and from different points of view (see [3], [4], [7], [9], [12], [16], [18], [23], [24] and literature, cited therein). Weighted composition operators are widely used in the description of asymptotic properties of dynamical systems and differential equations. A well-known example is provided by the celebrated Mather Theorem [19]. This theorem states that a diffeomorphism p of a finite dimensional smooth manifold X is Anosov (is hyperbolic, see the definition below) if and only if the associated weighted composition operator T is hyperbolic, that is r(T) fq qI’ 0 for the spectrum r (T) and unite circle ql". Here T acts in the space of continuous sections f of the tangent bundle over X, a is the differential of . This theorem was generalized in several directions (see [1], [2], [5], [14], [18]), and, in particular, for an arbitrary linear skew-product flow. To give the definition of the linear skew-product flow (LSPF) we consider a homeomorphism q of a compact metric space X and a continuous function a: X L(H) with values in the algebra L(H) of operators, bounded on a Hilbert space H. Let : X x Z+ ---> L (H) be a cocycle over q, defined by the rule (x, n) a(cp n- x)...., a(x). The linear skew-product flow, associated with , is the map (1) b’*: X x H X x H: (x,v) -> (dp’x, eP(x,n)v), n 6 Z+. The LSPFs are one of the major objects in studying the asymptotics of variational differential equations v’ A(dptx)v,x . X, where A: X L(H), and pt is a flow on X (see [10], [15], [21], [22] and the literature therein). One can thinkof (x, t) as the solving operator for the differential equation: v’(t) dp (x, t)v(O), ll, x X. One of the main problems here is the existence of exponential dichotomy (hy- perbolicity) for the LSPF (1) with continuous with respect to x dichotomy projec- tion (see [6], [8], [10], [15], [20], [21]). It means the existence of a continuous Received October 18, 1993 1991 Mathematics Subject Classification. Primary 47D06, 47B38; Secondary 34D20, 34G10. @ 1996 by the Board of Trustees of the University of Illinois Manufactured in the United States ofAmerica 21