Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 401-436

MEAN ERGODIC OPERATORS IN FRÉCHET SPACES

Angela A. Albanese, José Bonet and Werner J. Ricker

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

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

Katholische Universität Eichstätt-Ingolstadt, Mathematisch-Geographische Fakultät
D-85072 Eichstätt, Germany; werner.ricker 'at' ku-eichstaett.de

Abstract. Classical results of Pelczynski and of Zippin concerning bases in Banach spaces are extended to the Fréchet space setting, thus answering a question posed by Kalton almost 40 years ago. Equipped with these results, we prove that a Fréchet 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 Fréchet spaces are exhibited.

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

Key words: Mean ergodic operator, power bounded, Fréchet space, basis, Schauder decomposition.

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

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