Description of the interpolation spaces between \((\ell_ 1(\omega^ 0),\ell_ 1(\omega^ 1))\) and \((\ell_{\infty}(\omega^ 0),\ell_{\infty}(\omega^ 1))\).
Let E be a Banach ideal space of two-sided sequences on \({\mathbb{Z}}\). If we assume, that the shifts \(P_ k(a_ j)=(a_{k+j})^{\infty}_{j=- \infty} (k=0,\pm 1,...)\) are bounded operators in E, then \(\mu_ E=\lim_{k\to -\infty}\ln\| P_ k\| /k, \nu_ E=\lim_{k\to\infty }\ln\| P_ k\| /k.\) For every E and nonnegative function f on (0,\(\infty)\) let E(f) be the space with the norm \(\| (a_ j)\|_{E(f)}=\| (a_ jf(2^ j))\|_ E.\) Let \(f_ 0\), \(f_ 1\) be nonnegative concave functions on (0,\(\infty)\) with \(f_ 1/f_ 0\) increasing from 0 to \(\infty\). The author uses the following notations: \(h^{-1}=1/h, \mu^ i_ E=\mu_{E(f_ i)}, \nu^ i_ E=\nu_{E(f_ i^{-1})} (i=0,1)\), \(\overline{\ell_ p(f^{-1})}=(\ell_ p(f_ 0^{-1}),\ell_ p(f_ 1^{-1})).\) Theorem 1. The following statements are equivalent: 1) E is an interpolation space between the pairs \(\overline{\ell_ 1(f^{-1})} \overline{\ell_{\infty}(f^{-1})}; 2)\quad\mu^ 0_ E>0,\quad\nu^ 1_ E<1.\)
scientific article; zbMATH DE number 3875997