The splitting relation for Köthe spaces
The paper gives a complete characterization in terms of a condition on the defining matrices \(A=(a_{\nu,n})\), \(B=(b_{j,k})\) of those pairs (\(\lambda\) (A),\(\lambda\) (B)) of Köthe spaces, for which \(Ext^ 1(\lambda (A),\lambda (B))=0,\) i.e. every exact sequence \[ 0\to \lambda (A)\to F\to \lambda (B)\to 0 \] of Fréchet spaces splits. If \(\lambda (A)\not\cong \ell^ 1\) the condition can be stated as follows: \[ (S^*)\quad \forall \mu \exists n_ 0,k \forall K,m \exists n,S \forall \nu,j:\frac{a_{\nu,m}}{b_{j,k}}\leq S \max (\frac{a_{\nu,n}}{b_{j,K}},\frac{a_{\nu,n_ 0}}{b_{j,\mu}}). \] Remark: The first line in the proof of Prop. 1.5. is erroneous. Hence one has to assume \[ (S^*)_ 0\quad \forall \mu \exists n_ 0,k \forall K,m,R>0 \exists n,S \forall \nu,j:\frac{a_{\nu,m}}{b_{j,k}}\leq \max (S\frac{a_{\nu,n}}{b_{j,K}},\frac{1}{R}\frac{a_{\nu,n_ 0}}{b_{j,\mu}}) \] there and \(\lambda (A)\not\cong \ell^ 1\) in Theorem 1.1. By use of 1.5, 1.7, 1.2 and of \textit{A. E. Merzon}, Funkts. Anal. Prilozh. 6, 91-92 (1972; Zbl 0282.46002) one obtains: Theorem: \(Ext^ 1(\lambda (A),\lambda (B))=0\) if and only if (\(\lambda\) (A),\(\lambda\) (B)) satisfy \((S^*)_ 0.\) Condition \((S^*)\) is shown to be equivalent to a condition used by \textit{A. Grothendieck} for the characterization of topological properties of tensorproducts of (F)- and (DF)-spaces [Chap. II, {\S} 4 in Mem. Am. Math. Soc. 16 (1955; Zbl 0064.355)]. Taking up his investigation it is shown (assuming \(\Lambda\) (A) to be a Schwartz space) that \((S^*)\) is equivalent to each of the following properties: \(L_ b(\lambda (A),\lambda (B))\) is bornological (barrelled, quasibarrelled), the strong dual of \(L_ b(\lambda (A),\lambda (B))\) is quasicomplete.
scientific article; zbMATH DE number 3939964
scientific article