Positivity 13 (2009), 125–127
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010125-3, published online August 9, 2008
An extension of Sine’s counterexample
E. Yu. Emel’yanov and Nazife Erkursun
Abstract. We generalize Sine’s example  of a positive contraction in a
C(K)-space which is mean ergodic, but its square is not.
Mathematics Subject Classiﬁcation (2000). 46B42, 47A35.
Keywords. C(K)-space, positive contraction, isometry, mean ergodicity.
1. A bounded linear operator S in a Banach space X is called mean ergodic if the
of its Ces`aro averages A
converges strongly. It is
well known that S is mean ergodic if S
is mean ergodic for some m ∈ N (see for
example [1, Cor.2]). The converse is true for positive operators in ideally ordered
Banach spaces [2, Thm.2.1.9]. However, the converse is not true in general, even
for positive contractions. The ﬁrst example of such an operator is due to R. Sine,
who had constructed a positive isometry T in a C(K)-space, such that T is mean
ergodic, but T
is not [3, p.171]. Note that, if we omit the positivity assumption
on the operator, such an example can be constructed much more easily even in
C[0, 1] (cf. [1, p.206]).
2. We extend Sine’s construction and present a positive mean ergodic isometry
in a C(K)-space, such that its q-th power is not mean ergodic for an arbitrary
ﬁxed q ∈ N, q = 1. Consider the operator T ∈L(
(Z)) generated by the left shift
Then T is a positive invertible isometry in
(Z). Let 1 <p∈ N be a prime.
Deﬁne a bilateral sequence c
by the formula
and n = m + kp
and n = m + kp
be the closed subalgebra of
(Z) generated by the sequences T
all s ∈ Z and by the constant sequence (1)
. Obviously, C
is a Banach lattice