Multirectangular invariants for power Köthe spaces
This paper investigates the isomorphic classification of power Köthe spaces of first type \(E(\lambda,a)\) which are defined as follows: For sequences of positive numbers \(a=(a_i)_i\) and \(\lambda=(\lambda_i)_i\), \(E(\lambda,a)\) is the Köthe echelon space defined by the Köthe matrix \(A=(a_p)_p,\) \(a_p=(a_{i,p})_i\), \(a_{i,p}= \exp((-1/p + \lambda_i p)a_i)\), for each natural number \(i,p\). The class of all such spaces is the smallest class of Köthe echelon spaces, containing all Cartesian and projective tensor products of power series spaces and closed with respect to subspaces spanned by a subset of the canonical basis. Continuing previous work by Djakov and the present authors, the following result proven here solves an open problem in the area: Let \(X=E(\lambda,a)\) and \(Y=E(\mu,b)\) be two power Köthe spaces of first type which are isomorphic. If \(L\) is a basic subspace of \(X\) which is isomorphic to a power series space of finite (resp.\ infinite) type, then there is a basic subspace \(M\) of \(Y\) of the same kind which is quasidiagonally isomorphic to \(L\). Two Köthe echelon spaces \(\lambda(A)\) and \(\lambda(B)\) are quasidiagonally isomorphic if there is an isomorphism \(T\) from \(\lambda(A)\) onto \(\lambda(B)\) such that \(T e_i=t_i e_{\sigma(i)}\) for each \(i\) and for a sequence of scalars \((t_i)_i\) and a bijection \(\sigma\) on the natural numbers. The proof is long and technical. The crucial tools are improvements of the compound invariants which had been investigated before by the authors.
scientific article; zbMATH DE number 2107215
scientific article