We investigate ordered pairs of nudear K ~T H E spaces between which every linear continuous map is compact. We give some sufficient conditions for such pairs in terms of non-existence of corninon complemented subspaces. I n particular, in the case when one of the spaces is a weakly stable power series space, we obtain complete characterisations. Recently D. VOGT has obtained similar results [ 141. Our methods, being completely different from his, yield somewhat complementary results.
Preliminaries. Let E be a nuclear F R ~C ' H I G T (i.e. complete and metrisable) space (thbreviated NFS) over the field li (the field of real or complex numhers) with a btwis (r,,) whose coefficient functionnls are (I,,). lt is well known that all bases in 1 ' 3 are absolute: for ewh zero neighborliootl i I there exists a zero neighborhood V such that C Ift,(x)l pu(r,,) s p u ( z ) Vz€ E . Then it follows that /d is isomorphic to the sequence space n K b k J = {(fJ dP : II(Etb)llk = 2 ltnl ah-= +-V'k.1 n topologized by the seminorms (ll.llk), where a,, =yk(zn) with (pk} some increasing sequence of seminorms defining the topology of E. The matrix (ak,,) then satisfies in particular a k T , , r b a a k , b ~O V k , n (ii) V n sup aknw0 . k (0 Such matrices are called KOTHE matrices and the space K(a,) a K O T ~
space. Let a,bcb,denote (a,Jbn)€l-. We note that if two KOTHE matrices (aL.), (bk,J are equivalent, that is, V p 3 q a,,, 5 b, and V q 3 r b, <a, . , , , then the generated KOTHE spaces are the same, both as sets and as top.ologica1 spaces. Nuclearity of E is equivalent to the following condition on K(ah) : (iii) (GROTHENDIECK-PIETSCH Criterion) V p 3 q (am/aw),, E I , . If there exists a continuous norm on E then one can choose each seminorm p , to be a norm on E hence the condition (i') C Z , + ~, , , Z ~~= -O V k, n is also satisfied.
Throughout, E, P will denote nuclear KOTHE spaces with continuous norm and hence it will be assumed that the KOTHE matrices generating these spaces satisfy the conditions ( i ' ) , (ii), (iii) mentioned above.