TY - JOUR
T1 - Unimodular eigenvalues and weak mixing
AU - Jones, Lee K.
AU - Lin, Michael
N1 - Funding Information:
in part by the National
PY - 1980/1/1
Y1 - 1980/1/1
N2 - Let T be a linear operator on a Banach space X, with sup ∥ Tn ∥ < ∞. If {Tnx} is weak-* sequentially compact in X**, we prove the equivalence of the following: (1) limN → ∞ N-1 ∑n = 1N |〈 x*, Tnx〉| = 0 for every x*ε{lunate} X*. (2) x is orthogonal to the eigenvectors of T* with unimodular eigenvalues. For example, the assumption holds if X* is separable. An example shows that the sufficient condition for (1), Tninix → 0 weakly for some {ni}, is not necessary.
AB - Let T be a linear operator on a Banach space X, with sup ∥ Tn ∥ < ∞. If {Tnx} is weak-* sequentially compact in X**, we prove the equivalence of the following: (1) limN → ∞ N-1 ∑n = 1N |〈 x*, Tnx〉| = 0 for every x*ε{lunate} X*. (2) x is orthogonal to the eigenvectors of T* with unimodular eigenvalues. For example, the assumption holds if X* is separable. An example shows that the sufficient condition for (1), Tninix → 0 weakly for some {ni}, is not necessary.
UR - http://www.scopus.com/inward/record.url?scp=3142766509&partnerID=8YFLogxK
U2 - 10.1016/0022-1236(80)90079-8
DO - 10.1016/0022-1236(80)90079-8
M3 - Article
AN - SCOPUS:3142766509
SN - 0022-1236
VL - 35
SP - 42
EP - 48
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
ER -