Non-Schwartzian power series spaces
The author presents some isomorphism results for a class of countably normed power series spaces which are not necessarily nuclear or Schwartzian. Let \(a=(a_ n,n\geq 0,a_ n\uparrow \infty)\) be a sequence of numbers and H be a separable Hilbert space. Consider the spaces \(\Lambda_ 0(a;H)=\{x=(x_ k,k\geq 0),x_ k\in H:\| x\|^ 2_ p=\sum_{k}| x_ k|^ 2e^{-2a_ k/p}<\infty,\quad p\geq 1\}\) and \(\Lambda_{\infty}(a;H)\) is of the same structure except that the weights in the sum are \(e^{+2pa_ k}\) now. We can state the main Theorem. The following statements are equivalent: (1) \(\Lambda_ 0(a,H)\) is isomorphic to \(\Lambda_ 0(d,H)\) where \(d=(2^ k,k\geq 0),\) (2) the same as (1) with \(\Lambda_{\infty}\) replacing \(\Lambda_ 0.\) (3) The same as (1) with a satisfying \(\sup_{k}a_{k+1}/a_ k<\infty,\) (4) \(\Lambda_ 0(b,H)\) for some b satisfying \(\sup_{k}b_{k+1}/b_ k<\infty,\) is isomorphic to a closed complemented subspace of \(\Lambda_ 0(a,H).\) (5) The same as (4) with \(\Lambda_{\infty}\) in place of \(\Lambda_ 0.\) Some related consequences are also given.
scientific article; zbMATH DE number 3859738