Tameness of pairs of nuclear power series spaces and related topics

Abstract

The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios α m / β n , n , m ∈ N {\alpha _m}/{\beta _n},n,m \in {\mathbf {N}} , is bounded, (ii) Every operator from Λ ∞ ( β ) {\Lambda _\infty }(\beta ) to Λ 1 ( α ) {\Lambda _1}(\alpha ) is compact, (iii) The pair ( Λ ∞ ( β ) , Λ 1 ( α ) ) ({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha )) is tame, i.e., for every operator T T from Λ ∞ ( β ) {\Lambda _\infty }(\beta ) to Λ 1 ( α ) {\Lambda _1}(\alpha ) there is a positive integer a a such that for every k ∈ N k \in {\mathbf {N}} there is a constant C k {C_k} such that | | T x | | k ⩽ C k | x | a k ||Tx|{|_k} \leqslant {C_k}|x{|_{ak}} for every x ∈ Λ ∞ ( β ) x \in {\Lambda _\infty }(\beta ) . (iv) Every short exact sequence 0 → Λ τ ( β ) → X → Λ 1 ( α ) → 0 0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0 , where τ = 1 \tau = 1 or ∞ \infty , splits. (v) Λ 1 ( α ) × Λ ∞ ( β ) {\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta ) has a regular basis, (vi) Λ 1 ( α ) ⊗ Λ ∞ ( β ) {\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta ) has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.