Ergodic and Spectral Analysis of Certain Infinite Measure Preserving Transformations

0. Introduction. Throughout this paper T will denote a measure preserving transformation on a cr-finite infinite measure space (X, (B, m) which is point isomorphic with the Lebesgue measure space of the real line. Unless otherwise stated, T will be one-one. Equations involving functions or sets will always be interpreted modulo sets of measure zero. T is said to be ergodic if T~1E = E, ££(B, implies either mE = 0 or m (X-E)=0. P. R. Halmos [l ] has posed the problem of characterising spectrally the property of ergodicity for cr-finite infinite measure preserving transformations. Our purpose is to show that the ergodicity of such a transformation is not a spectral property. Let Ut denote the induced unitary operator of L2(X,(S>,m) onto itself defined by