A decomposition of weighted translation operators

Abstract

Let (X, A, m) be a measure space and T an invertible measure-preserving transformation on X. Given ϕ \phi in L ∞ ( X ) {L^\infty }(X) , define operators M ϕ {M_\phi } and U T {U_T} on L 2 ( X ) {L^2}(X) by ( M ϕ f ) ( x ) = ϕ ( x ) f ( x ) ({M_\phi }f)(x) = \phi (x)f(x) and ( U T f ) = f ( T x ) ({U_T}f) = f(Tx) . Operators of the form M ϕ U T {M_\phi }{U_T} are called weighted translation operators. In this paper it is shown that every weighted translation operator on a sufficiently regular measure space an be decomposed into a direct integral of weighted translation operators where almost all of the transformations in the integrand are ergodic. It is also shown that every hyponormal weighted translation operator defined by an ergodic transformation is either normal or unitarily equivalent to a bilateral weighted shift. These two results along with some results concerning direct integrals of hyponormal and subnormal operators are used to show that every hyponormal (resp. subnormal) weighted translation operator is unitarily equivalent to a direct integral of normal operators and hyponormal (resp. subnormal) bilateral weighted shifts. The paper concludes with an example.