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On automatic boundedness of Nemytskiĭ set-valued operators

S. Rolewicz, Wen Song (1995)

Studia Mathematica

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let N F be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function F : Ω × X → 2 Y . It is shown that if N F maps a modular space ( N ( L ( Ω , Σ , μ ; X ) ) , ϱ N , μ ) into subsets of a modular space ( M ( L ( Ω , Σ , μ ; Y ) ) , ϱ M , μ ) , then N F is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that r K = s u p ϱ N , μ ( x ) : x ∈ K < ∞ we have s u p ϱ M , μ ( y ) : y ∈ N F ( K ) < ∞ .

On exact sequences of quojections.

G. Metafune, V. B. Moscatelli (1991)

Revista Matemática de la Universidad Complutense de Madrid

We give some general exact sequences for quojections from which many interesting representation results for standard twisted quojections can be deduced. Then the methods are also generalized to the case of nuclear Fréchet spaces.

On non-primary Fréchet Schwartz spaces

J. Díaz (1997)

Studia Mathematica

Let E be a Fréchet Schwartz space with a continuous norm and with a finite-dimensional decomposition, and let F be any infinite-dimensional subspace of E. It is proved that E can be written as G ⨁ H where G and H do not contain any subspace isomorphic to F. In particular, E is not primary. If the subspace F is not normable then the statement holds for other quasinormable Fréchet spaces, e.g., if E is a quasinormable and locally normable Köthe sequence space, or if E is a space of holomorphic functions...

On the algebra of smooth operators

Tomasz Ciaś (2013)

Studia Mathematica

Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fréchet algebra L(s',s) of so-called smooth operators. We also characterize closed commutative *-subalgebras of L(s',s) and establish a Hölder continuous functional calculus in this algebra. The key tool is the property (DN) of s.

On the derived tensor product functors for (DF)- and Fréchet spaces

Oğuz Varol (2007)

Studia Mathematica

For a (DF)-space E and a tensor norm α we investigate the derivatives T o r α l ( E , · ) of the tensor product functor E ⊗ ̃ α · : → from the category of Fréchet spaces to the category of linear spaces. Necessary and sufficient conditions for the vanishing of T o r ¹ α ( E , F ) , which is strongly related to the exactness of tensored sequences, are presented and characterizations in the nuclear and (co-)echelon cases are given.

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