TY - CHAP

T1 - Poisson’s equation for mean ergodic operators

AU - Lin, Michael

AU - Suciu, Laurian

N1 - Publisher Copyright:
© 2015 M. Lin, L. Suciu.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Let T be a bounded linear operator on a Banach space χ. For T power-bounded, mean ergodicity and weak mean ergodicity are equivalent, but in general, even in Hilbert spaces, this is not so. For T weakly mean ergodic, Poisson’s equation (I − T)x = y can be solved for a given y if and only if (formula presented) converges weakly. In this paper we study, for T weakly mean ergodic, the set S(T) := {y ∈ X : supn ||An(T)y|| < ∞}. We show that for T Cesàro bounded with Tn/n → 0 weakly, S(T) is closed if and only if (I −T)χ is closed, and then T is weakly mean ergodic; if ||Tn||/n → 0, then T is uniformly ergodic. As an application of our study in reflexive spaces, we supplement the Fonf-Lin-Wojtaszczyk characterization of reflexivity of Banach spaces with a basis: Such a space χ is reflexive if and only if every weakly mean ergodic T satisfies (I − T)χ = S(T), if and only if every power-bounded mean ergodic T satisfies (I − T)7 = S(T).

AB - Let T be a bounded linear operator on a Banach space χ. For T power-bounded, mean ergodicity and weak mean ergodicity are equivalent, but in general, even in Hilbert spaces, this is not so. For T weakly mean ergodic, Poisson’s equation (I − T)x = y can be solved for a given y if and only if (formula presented) converges weakly. In this paper we study, for T weakly mean ergodic, the set S(T) := {y ∈ X : supn ||An(T)y|| < ∞}. We show that for T Cesàro bounded with Tn/n → 0 weakly, S(T) is closed if and only if (I −T)χ is closed, and then T is weakly mean ergodic; if ||Tn||/n → 0, then T is uniformly ergodic. As an application of our study in reflexive spaces, we supplement the Fonf-Lin-Wojtaszczyk characterization of reflexivity of Banach spaces with a basis: Such a space χ is reflexive if and only if every weakly mean ergodic T satisfies (I − T)χ = S(T), if and only if every power-bounded mean ergodic T satisfies (I − T)7 = S(T).

UR - http://www.scopus.com/inward/record.url?scp=85106790385&partnerID=8YFLogxK

U2 - 10.1090/conm/636/12733

DO - 10.1090/conm/636/12733

M3 - Chapter

AN - SCOPUS:85106790385

T3 - Contemporary Mathematics

SP - 141

EP - 148

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -