Banach Spaces With Unique Unconditional Basis, Up to by J. Bourgain, P. G. Casazza, J. Lindenstrauss

By J. Bourgain, P. G. Casazza, J. Lindenstrauss

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J ^ ' "" 10 if ; & Ni and s ^ A . , otherwise We ihave t k a t ^S^H ^ 1, for all t, an

1. Let \qn\£=1 integers be an increasing T^\ sequence and, for each n , let yn e [^]/=g^+i be so that \\yn\\(p) for some fixed p ^ 1. Then, for any choice of scalars integers {kn \£=i with q^ + l^k^ 3 " 1 ' * II t n=l of = 1, [ a n J * = 1 and ^ ^ n + i . 1 is as follows. 1'. 2. Let p > 1 and Ex < E2 < ' ' ' be finite subsets of the integers. For each n, let L^ be a linear operator from EnT^ into itself Then the operator L : 7 ^ -» 7 ^ , defined by oo x x € to - ZI Ln,En i ^ ^ ^ bounded if and only if sup ||Z

I is equivalent to Je^J^!. If, on the other hand, tp restricted to pairs of integers selected from Nx is equal to zero then \m;m ^ * ( p ), w m (fc(l)) > ir m (*(2)) > for any choice of p. Thus, as above, (2C)- 1 !! t

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