Power Series Subspaces Of Nuclear Frechet Spaces With The Properties DN And Omega

Abstract

Power series spaces constitute an important and well-studied class in the theory of Fréchet spaces. Linear topological invariants weak-DN and Ω are enjoyed by many natural Fréchet spaces appearing in analysis. In particular, spaces of analytic functions, solutions of homogeneous elliptic linear partial differential operator with their natural topologies have the properties weak-DN and Ω. It is a well-known fact that the diametral dimension ∆(E) and the approximate diametral dimension δ(E) of a nuclear Fréchet space E with the properties weak-DN and Ω are between corresponding invariant of power series spaces Λ_{1}(ε) and Λ_{∞}(ε) for some specific exponent sequence ε. This sequence is called associated exponent sequence of E, see Definition 2.4.2. Concidence of the diametral dimension and/or approximate diametral dimension of E with that of a power series space yields some structural results. For example, in [1], A. Aytuna, J.Krone and T. Terzioglu proved that a nuclear Fréchet space E with the properties weak-DN and Ω contains a complemented copy of Λ_{∞}(ε) provided ∆(E) = ∆(Λ_{∞}(ε)) and ε is stable. On the other hand, A. Aytuna, [2], characterized tame nuclear Fréchet spaces E with the properties weak-DN and Ω and stable exponent sequence ε, as those that satisfies δ(E) = δ(Λ_{1}(ε)). These results lead us to ask the following two questions: Let E be a nuclear Fréchet space with the properties weak-DN and Ω and ε be the associated exponent sequence of E. 1. Is there a complemented subspace of E which is isomorphic to Λ_{1}(ε) if ∆(E) = ∆(Λ_{1}(ε))? 2. If the diametral dimension of E coincides with that of a power series space, then does this imply that the approximate diametral dimension also do the same and vice versa? The basis of this thesis was motivated by these two questions. The main purpose of this thesis is to determine the connections between the diametral dimension and the approximate diametral dimension and to investigate power series subspaces of the nuclear Fréchet spaces with the properties weak-DN and Ω using these invariants. In the first chapter, some significant studies in the theory of nuclear Fréchet spaces are mentioned and the aim of this thesis is given. In the second chapter, we introduced preliminary materials and essential theorems. In the third chapter, we showed that the second question has an affirmative answer when the power series space is of infinite type. Then we searched an answer for the second question in the finite type case and, in this regard, we first proved that the condition δ (E) = δ (Λ_{1}(ε)) always implies ∆(E) = ∆(Λ_{1}(ε)). For other direction, the existence of a prominent bounded subset in the nuclear Fréchet space E plays a decisive role. Among other things, we proved that δ (E) = δ (Λ_{1}(ε)) if and only if E has a prominent bounded subset and ∆(E) = ∆(Λ_{1} (ε)). In the first section of the fourth chapter, we showed that a regular nuclear Köthe space with the properties DN and Ω is a power series space if its diametral dimension coincides with that of a power series space of infinite type or its approximate diametral dimension coincides with that of a power series space of finite type. In the second section of the fourth chapter, we constructed a family of nuclear Köthe spaces K(a_{k,n}) with the properties weak-DN and Ω. First we showed that for an element of the family of which is parameterized by a stable sequence α, ∆(K(a_{k,n})) = ∆(Λ_{1}(α)) and δ(K(a_{k,n})) = δ(Λ1(α)). Second, we proved that for an element of the family of which is parameterized by an unstable sequence α, ∆(K(a_{k,n}) = ∆(Λ_{1}(ε)) and δ(K(a_{k,n}))≠δ(Λ1(ε)) for its associated exponent sequence ε. This showed that the second question has a negative answer for power series space of finite type. Furthermore, we proved in Theorem 4.3.1 that the first question has a negative answer, that is, Λ1(ε) is not isomorphic to any subspace of these Köthe spaces K(a_{k,n}), let alone is isomorphic to a complemented subspace, though the condition ∆(K(a_{k,n})) = ∆(Λ_{1}(ε)) is satisfied. In the third section of fourth chapter, motivated by our finding in the third section, we compiled some additional information, for instance, for an element E of the family parameterized by an unstable sequence, • E does not have a prominent bounded set. • ∆(E), with respect to the canonical topology, is not barrelled, hence, not ultrabornological. • Although the equality ∆(E) = Λ_{1}(ε) is satisfied and the canonical imbedding from ∆(E) into Λ_{1}(ε) has a closed graph, the canonical imbedding from ∆(E) into Λ1(ε) is not continuous.