Universal generators for varieties of nuclear spaces

Abstract

It is shown that a product of several copies of Λ ( β ϕ ) \Lambda ({\beta ^\phi }) is a universal ϕ \phi -nuclear space if the power series space Λ ( β ϕ ) \Lambda ({\beta ^\phi }) with β k ϕ = − log ⁡ ( ϕ − 1 ( 1 / k + 1 ) ) , k ϵ { 0 , 1 , 2 , ⋯ } \beta _k^\phi = - \log ({\phi ^{ - 1}}(1/\sqrt {k + 1} )),k\;\epsilon \;\{ 0,1,2, \cdots \} , is ϕ \phi -nuclear; here ϕ = [ 0 , ∞ ) → [ 0 , ∞ ) \phi = [0,\infty ) \to [0,\infty ) is a continuous, strictly increasing subadditive function with ϕ ( 0 ) = 0 \phi (0) = 0 . In case Λ ( β ϕ ) \Lambda ({\beta ^\phi }) is not ϕ \phi -nuclear the sequence space Λ ( l ϕ + ) \Lambda (l_\phi ^ + ) is a ϕ \phi -nuclear space with the property that every ϕ \phi -nuclear space is isomorphic to a subspace of a product of Λ ( l ϕ + ) \Lambda (l_\phi ^ + ) if lim sup t → 0 ( ϕ ( t ) ) − 1 ϕ ( t ) > ∞ {\lim \;\sup _{t \to 0}}{(\phi (t))^{ - 1}}\phi (\sqrt t ) > \infty .