408 XV NORMED ALGEBRAS AND SPECTRAL THEORY

16. Let X be a locally compact space, p a bounded positive measure on X with total mass
1, and u a /z-measurable mapping of X into X such that /x is invariant with respect to
u (Section 13.9, Problem 24). Let U be the unitary operator on L£(X, /x) such that
£/./=(/oW)~ (Section 13.11, Problem 10).

(a) The mapping u is said to be mixing (resp. weakly mixing ) with respect to /LI if,
for each pair of /^-measurable subsets A, B of X we have

lim (M«-fl(A) n B)) = /i(A)/*(B)

11-* 00

(resp.

lim - "l |/*(ir *CA) n B) - /*(A)/«(B)| = 0.)

n-*o> n ks;Q

Every mixing mapping is weakly mixing. Every weakly mixing mapping is ergodic
(Section 13.9, Problem 13(d)).

(b) Show that u is mixing (resp. weakly mixing) if and only if, for each pair of
functions/, g in ^c(^» A0» we nave

(resp.

An equivalent condition is that, for each / e .#c(X» p) such that (/11) = 0
(i.e., If dp, = 0), we have

lim 0/M-/I/) = 0

n-»oo

(resp.

lim I "SKI/" -f\S) - (/I D(l \9) I = 0).

(Replace / by /+ g, and remark that if a sequence (<?„) satisfies the condition

•1 »~1 1 n~1

lim - ^ \ak\2 = 0, then also lim - ]£ \ak\ = 0, by the Cauchy-Schwarz inequality.)

n-*oo n IcssO «-*« n k*=Q

(c) For u to be ergodic with respect to /x, it is necessary and sufficient that 1
should be an eigenvalue of U with multiplicity 1. If this is so, then all the eigen-
values of U have multiplicity 1 and form a subgroup of the group U of complex
numbers of absolute value 1. For each eigenvector /e Lc(X, (JL) of £/, the function
|/| is constant almost everywhere. (Remark that if £/-/=A/and U - g = \g, then

(d) Show that the following properties are equivalent :

(a) u is weakly mixing with respect to /it.

(/?) u x u is an ergodic mapping of X x X into X x X with respect to the
measure /u, (g) /x.

(y) The only eigenvalue of U is 1.

(To show that (a) implies (/?), consider subsets M x N of X x X, where M and N
are /^-measurable subsets of X. To prove that (/8) implies (y), observe that if /is and K a compact