Ergodic and spectral analysis of certain infinite measure preserving transformations

T is said to be ergodic if T-1E=E, ECG3, implies either mE=O or m (X-E)=O. P. R. Halmos [1 ] has posed the problem of characterising spectrally the property of ergodicity for ar-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,63,m) onto itself defined by