On tame pairs of Fréchet spaces

Abstract

We characterize tame pairs (X, Y ) of Fréchet spaces where either X or Y is a power series space. For power series spaces of finite type, we get the well-known conditions of (DN)-(Ω) type. On the other hand, for power series spaces of infinite type, surprisingly, tameness implies boundedness of every linear and continuous operator. Next, we prove that every tame Fréchet space is quasi-normable. This result extends earlier result of the author valid only for Köthe sequence spaces.

Tameness in Fréchet spaces of analytic functions

“…Hence it follows that X , Λ 1 (E 2n ) ∈ T . Now Theorem 11 of [17] implies that X satisfies the strong Ω condition,Ω, of Vogt. This together with our assumption that, X has the property DN allows us to conclude that X is isomorphic to a finite type power series space (Proposition 2.9.18 of [15]).…”

“…This structure allows one to use the results of well studied LF-spaces in the study of L(X , Y) for (X , Y) ∈ T . These ideas are used in the study of nuclear Fréchet spaces X which form a tame pair with nuclear stable power series spaces of finite or infinite type in [17] where a complete characterization of such spaces in terms of the linear topological invariants of Vogt are obtained.…”

(DN)-(Ω)-type conditions for Fréchet operator spaces

“…They play a role in characterizing when L(X, Y ) = L B(X, Y ) that is, when every linear and continuous operator between Fréchet spaces is bounded in the sense that it maps some zero neighborhood into a bounded set; see [Meise and Vogt 1997, Chapter 29;Vogt 1983]. These conditions appear also in the lately defined concept of tameness; see [Dubinsky and Vogt 1989;Piszczek 2009]. Both boundedness and tameness are strongly connected with the longstanding open problem of Pełczyński of whether every complemented subspace of a nuclear Fréchet space with a basis has a basis itself.…”