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Resumen:
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The work under consideration fits into the following general circle of problems: Given a Banach space B which possesses a distinguished basis (ei) i?N and a bounded linear operator A:B?C, to what extent does the sequence (Aei) constitute some sort of basis (ai) on C, where ai=Aei? It turns out to be more suitable to work with systems of rays (ri)i?N (that is, one-dimensional subspaces) such that ai?ri. Compact operators A are excluded for what turn out to be obvious reasons, and the operators A are required to be injective. This leads at various points to a consideration of cases: the range R(A) of A is a closed subspace; and R(A) is dense in C but not equal to C. The paper is devoted principally to the case B=C=l2 with the distinguished basis (ei) being a complete orthonormal set (c.o.s.). There are also results applying to lp, p?2. A sequence (ai) is said to be doubly bounded (d.b.) provided that 00 for some particular c.o.s. (e?i) contained in a previously fixed linear subspace dense in l 2. Henceforth A represents a noncompact injective operator with R(A)?R(A) ¯ ¯ ¯ . It is proved that there exists a c.o.s. (e?i) such that the L-system (Ae?i) is d.b., complete in R(A) ¯ ¯ ¯ and heterogonal in blocks. Furthermore, it is shown that for a noncompact A, the following are equivalent: (i) N(A)=0. (ii) There exists a c.o.s. (ei) such that (Aei) is a strong M base which is d.b. in R(A) ¯ ¯ ¯ . Definition: A sequence (ai) is minimal if ai is not in the closed linear subspace spanned by the aj, j?i. An M-base is a complete minimal sequence (ai) such that ? ? i=1 [a i ,a i+1 ,?]=0 where [a i ,a i+1 ?] represents the closed linear subspace spanned by ai, a i+1,?. A further theorem states that given a sequence of rays (ri) the following are equivalent: (I) (ri) is an SLR; (II) for (ai?ri?{0}) one has ? ? i=1||ai||2 if and only if (ai) is summable. A final section of the paper is devoted to lp, p?2. It is shown here that if p>2 and (xn) is a d.b. sequence in lp then the following are equivalent: (1) (xn) is weakly p-summable; (2) ? ? 1 ? n x n converges unconditionally if and only if ? ? 1 |? n | p? , where 1/p+1/p? =1. For p
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