In 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 Mbasic sequences.
PLANS and GARC~A CASTELL~N have characterized the boundedneas of a linear operator T by requiring the minimality of any sequence whose image is a minimal sequence (e. g. [P, 19691, [GC, 19901). We extend these results to other types of requences like M-basic, basic, rtrong M-basic, ctc.. 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 pminimal sequence into q -minimal.
In the last section we deal with M -basis whose image is Mbasis or norming Mbasis or basis or in general the "best" possible eequence.
1. Notation and definitions
Let X, Y, . . . be separable infinite dimensional Banach spaces and X', Y * , . . . be their topological dual spaces.
Let J = {a, : n E IN} be a sequence in X. We denote by Ws or [ak : k E S ] the smallest closed linear subspace spanned by {ar, : k E S}, and by (ak : k E S) the smallest linear subspace spanned by {ar, : k € s}. We call the set {Ws : S C IN} the ge ometry of J, denoted by G ( J ) . If X = [a, : n E IN], then {a, : n E IN} is called complete or fundamental. The kernel of J is defined by 1991 Mothemaiics Subject Claasificotion. Primary 46 B 99; Secondary 46 B 20, 47 A05. Keywords and phmaes. sequences, kernel m d m g e of M operator.