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On the equivalence of certain sets of sequences. (English) Zbl 0584.46005

Let s\( \)be the set of all infinite complex sequences, \(p=(p_ k)\) a sequence of strictly positive numbers and \(t=(t_ k):=(p_ k)\). Let \(A\), \(B\) denote the following two classes of subsets of \(s\): \[ \begin{aligned} c_ 0(p) &:= \left\{x\in s\left| | x_ k|^{p_ k}\right. =o(1)\right\}\tag{Class A:}\\ \ell_{\infty}(p) &:=\left\{x\in s\left|| x_ k|^{p_ k}\right. =0(1)\right\} \\ \ell (p) &:= \left\{x\in s\left| \Sigma_ k| x_ k|^{p_ k}\right. <\infty\right\}. \\ c_ 0\{p\} &:= \left\{x\in s| \exists r>0:| x_ kr|^{p_ k}t _ k=o(1)\right\} \tag{Class B:}\\ \ell_{\infty}\{p\} &:= \left\{x\in s| \exists r>0:| x_ kr|^{p_ k}t _ k=0(1)\right\} \\ \ell \{p\} &:= \left\{x\in s| \exists r>0:\Sigma_ k| x_ kr|^{p_ k} t_ k<\infty \right\}. \end{aligned} \] Finally, let \(c_ 0,\ell_\infty\) and \(\ell_1\) denote the spaces of null, bounded and absolutely summable sequences, respectively.
Two sets of sequences \(R,Q\subseteq s\) are said to be equivalent (in the sense of Nakano) if there exists a strictly positive sequence \(u=(u_ k)\) such that the mapping \[ u:R\to Q, \quad x\in R\to y:=ux:=(u_ kx_ k)\in Q \] is a one-to-one correspondence between \(R\) and \(Q\). This fact is denoted by \(R\simeq Q(u)\) or simply \(R\simeq Q\). The sequence u is not necessarily unique. The problem of determining the sequence \(u\) for which \(R\simeq Q(u)\) is closely related to the problem of characterizing the class D(R,Q) of the diagonal matrix transformations between R and Q.
In this paper we characterize certain diagonal matrix transformations and obtain some results concerning inclusion relations and equivalence between sets from classes A and B. We also prove that although \(c_ o(p)\) has the Schur property whenever \(p\in c_ o\), there does not exist \(p\in c_ o\) for which \(c_ o(p)\simeq \ell_ 1\).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46A04 Locally convex Fréchet spaces and (DF)-spaces
46B25 Classical Banach spaces in the general theory
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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