Annales Academi� Scientiarum Fennic�
Mathematica
Volumen 34, 2009, 401-436

MEAN ERGODIC OPERATORS IN FR�CHET SPACES

Angela A. Albanese, Jos� Bonet and Werner J. Ricker

Universit� del Salento, Dipartimento di Matematica ``Ennio De Giorgi''
C.P. 193, I-73100 Lecce, Italy; angela.albanese 'at' unile.it

Universidad Polit�cnica de Valencia, Instituto Universitario de Matem�tica Pura y Aplicada
Edificio IDI5 (8E), Cubo F, Cuarta Planta, E-46071 Valencia, Spain; jbonet 'at' mat.upv.es

Katholische Universit�t Eichst�tt-Ingolstadt, Mathematisch-Geographische Fakult�t
D-85072 Eichst�tt, Germany; werner.ricker 'at' ku-eichstaett.de

Abstract. Classical results of Pelczynski and of Zippin concerning bases in Banach spaces are extended to the Fr�chet space setting, thus answering a question posed by Kalton almost 40 years ago. Equipped with these results, we prove that a Fr�chet space with a basis is reflexive (resp. Montel) if and only if every power bounded operator is mean ergodic (resp. uniformly mean ergodic). New techniques are developed and many examples in classical Fr�chet spaces are exhibited.

2000 Mathematics Subject Classification: Primary 46A04, 46A35, 47A35; Secondary 46G10.

Key words: Mean ergodic operator, power bounded, Fr�chet space, basis, Schauder decomposition.

Reference to this article: A.A. Albanese, J. Bonet and W.J. Ricker: Mean ergodic operators in Fr�chet spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 401-436.

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