AbstractIn this paper we study the action of a bounded linear operator over different kinds of sequences of a Banach space. Our work is mainly devoted to minimal and M ‐ basic sequences.Plans and García Castellón have characterized the boundedness of a linear operatorTby requiring the minimality of any sequence whose image is a minimal sequence (e.g. P, 1969, GC, 1990). We extend these results to other types of sequences like M‐basic, basic, strong M‐basic, etc.,We are also interested on conditions that ensure the minimality of the image of a given minimal sequence. Thus in Corollary 3.7 we characterize semi ‐ Fredholm operators as those which transform every p‐minimal sequence into q‐minimal.In the last section we deal with M ‐ basis whose image is M ‐ basis or norming M ‐ basis or basis or in general the “be
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