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Ergodic theory is a relatively new branch of mathematics which from a mathematical point of view may be regarded as generated by the interaction of measure theory and the theory of transformation groups. Its basic concept of "metric transitivity" or "ergodicity" was introduced in 1928 in a paper of Paul Smith and G. D. Birkhoff on dynamical systems. However, the significance of this concept was not appreciated until late 1931 when J. yon Neumann and G. D. Birkhoff proved the celebrated mean and pointwise ergodic theorems, and one may regard the nearly simultaneous appearance of these papers as marking the birth of the subject. Birkhoff's proof of the much more difficult pointwise ergodic theorem was stimulated by yon Neumann's theorem and yon Neumann, in turn, was stimulated by a key observation of B. O. Koopman. Let De be a surface of constant energy E in the phase space D of some Hamiltonian dynamical system. Let V,(~o) denote the point of phase space representing the "state" of the system t time units after it was represented by ~o. Then, for each t, oJ -. Vt(oJ ) is a one-to-one transformation of f2 e onto itself which conserves the natural volume element ~e in f2e induced in £2 e by the Liouville measure dql .." dqn dpl "'" d p n . Moreover, Vq+t~ = VqVt~ for all real numbers t 1 and tz. Koopman's observation (not so obvious 40 years ago as now) was that we may obtain a unitary representation't --+ Ut of the additive group of the real line in the Hilbert space 5°2(f2e, ~e) by defining Ut( f ) (co ) = f (V, ( , -o) ) .

In this paper we give a general expression for operators on a Hilbert space which are similar to an operator of a certain type such as a contraction, a unitary operator, a dissipative operator, an operator with nuclear imaginary component, etc. Theorem 1 covers these types and many others. Theorem 2 is concerned with those operators whose spectrum lies in the open upper half-plane. By an operator we always mean a bounded linear operator on a Hilbert space !$. As usual H > 0 means that H is a positive operator, i.e., (Hx, x) 2 0 f or all x E &. We write H > 0 to indicate that H is positive and invertible. The spectrum of an operator B is denoted bY 4B).

It is shown in this paper (Theorem 1) that if α and β are real numbers such that α gb is irrational and algebraic, then there exist two (necessarily distinct) distributions S and T on R, both with compact supports, such that δ′ = ΔαS + ΔβT. Here ΔαS means S − Sα and Sα denotes the translate of S by α. It is also shown that δ′ has no such representation if α gb has rational or certain transcendental values, and that S and T can be chosen to have order two and no lower order. If ϑ belongs to any of the spaces D, E, S or their duals then ϑ ∗ S and ϑ ∗ T belong to the same space as ϑ, and ϑ′ = Δ α (ϑ ∗ S) + Δ β (ϑ ∗ T) . The formula δ′ = ΔαS + ΔβT is generalized to Rn by means of the tensor product of distributions, and it follows from this formula that there is no discontinuous translation-invariant linear form on any of the spaces D (Rn), E (Rn), S (Rn) or their duals. The same thing is also proved for E (Tn) and its dual where Tn denotes the n-dimensional torus group.