Publication: Sistemas L de rayos y sumabilidad
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1986
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Universidad Complutense de Madrid
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Abstract
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 0<infi||ai||≤supi ||ai||<∞. A sequence (ai) is called an L-system if ai=Aei, i∈N, where A is a bounded operator and (ei) is a c.o.s. A system (ri) of rays (SLR) is one such that there exists a d.b. L-system (ai) such that ai∈ri. An SLR (ri) cannot have a ray of weak accumulation. Since for a compact operator A we have Aei ⇀0 (weak convergence) with (ei) a c.o.s., no compact A gives rise to an associated SLR sequence. One has the following cases for a noncompact operator A with trivial null-space (N(A)=0) : (a) A has a bounded left-inverse A −1. Then all its L-systems subtend SLR and are heterogonal. (b) A −1 is not bounded. It is this case which receives attention now. An operator A with N(A)=0 is noncompact if and only if infi∥Ae'i||>0 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 ξ n x n converges unconditionally if and only if ∑ ∞ 1 |ξ n | p′ <∞, where 1/p+1/p′ =1. For p<2, there are no d.b. weakly summable sequences in lp.