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A Riesz representation theory for completely regular Hausdorff spaces and its applications

Marian Nowak — 2016

Open Mathematics

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we...

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak — 2005

Banach Center Publications

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space ( X , | | · | | X ) let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function | | f ( · ) | | X belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let D u ( = f E ( X ) : | | f ( · ) | | X u ) stand for the order interval in E(X). For a real Banach space ( Y , | | · | | Y ) a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈ E⁺ the set...

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak — 2011

Banach Center Publications

Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let L Φ ( X ) be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to L Φ ( X ) . In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to L ( X ) is ( τ ( L ( X ) , L ¹ ( X * ) ) , | | · | | Y ) -compact.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak — 2011

Banach Center Publications

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let i : L ( X ) L ¹ ( X ) stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then T i : L ( X ) Y is a weakly compact operator. Moreover, we obtain that if T: L¹(X)...

Topological properties of some spaces of continuous operators

Marian Nowak — 2016

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C b ( X , E ) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space L β ( C b ( X , E ) , F ) of all ( β , | | · | | F ) -continuous linear operators from C b ( X , E ) to F, equipped with the topology τ s of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize τ s -compact subsets of L β ( C b ( X , E ) , F ) in terms of properties of the corresponding sets of the representing...

Strong topologies on vector-valued function spaces

Marian Nowak — 2000

Czechoslovak Mathematical Journal

Let ( X , · X ) be a real Banach space and let E be an ideal of L 0 over a σ -finite measure space ( Ø , Σ , μ ) . Let ( X ) be the space of all strongly Σ -measurable functions f Ø X such that the scalar function f ˜ , defined by f ˜ ( ø ) = f ( ø ) X for ø Ø , belongs to E . The paper deals with strong topologies on E ( X ) . In particular, the strong topology β ( E ( X ) , E ( X ) n ) ( E ( X ) n = the order continuous dual of E ( X ) ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

Order continuous seminorms and weak compactness in Orlicz spaces.

Marian Nowak — 1993

Collectanea Mathematica

Let L-phi be an Orlicz space defined by a Young function phi over a sigma-finite measure space, and let phi* denote the complementary function in the sense of Young. We give a characterization of the Mackey topology tau(L*,L-phi*) in terms of some family of norms defined by some regular Young functions. Next we describe order continuous (=absolutely continuous) Riesz seminorms on L-phi, and obtain a criterion for relative sigma(L-phi,L-phi*)-compactness in L-phi. As an application we get a representation...

Strict topologies and Banach-Steinhaus type theorems

Marian Nowak — 2009

Commentationes Mathematicae Universitatis Carolinae

Let X be a completely regular Hausdorff space, E a real Banach space, and let C b ( X , E ) be the space of all E -valued bounded continuous functions on X . We study linear operators from C b ( X , E ) endowed with the strict topologies β z ( z = σ , τ , , g ) to a real Banach space ( Y , · Y ) . In particular, we derive Banach-Steinhaus type theorems for ( β z , · Y ) continuous linear operators from C b ( X , E ) to Y . Moreover, we study σ -additive and τ -additive operators from C b ( X , E ) to Y .

Inductive limit topologies on Orlicz spaces

Marian Nowak — 1991

Commentationes Mathematicae Universitatis Carolinae

Let L ϕ be an Orlicz space defined by a convex Orlicz function ϕ and let E ϕ be the space of finite elements in L ϕ (= the ideal of all elements of order continuous norm). We show that the usual norm topology 𝒯 ϕ on L ϕ restricted to E ϕ can be obtained as an inductive limit topology with respect to some family of other Orlicz spaces. As an application we obtain a characterization of continuity of linear operators defined on E ϕ .

Topological dual of non-locally convex Orlicz-Bochner spaces

Marian Nowak — 1999

Commentationes Mathematicae Universitatis Carolinae

Let L ϕ ( X ) be an Orlicz-Bochner space defined by an Orlicz function ϕ taking only finite values (not necessarily convex) over a σ -finite atomless measure space. It is proved that the topological dual L ϕ ( X ) * of L ϕ ( X ) can be represented in the form: L ϕ ( X ) * = L ϕ ( X ) n L ϕ ( X ) s , where L ϕ ( X ) n and L ϕ ( X ) s denote the order continuous dual and the singular dual of L ϕ ( X ) respectively. The spaces L ϕ ( X ) * , L ϕ ( X ) n and L ϕ ( X ) s are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory...

Bochner representable operators on Köthe-Bochner spaces

Marian Nowak — 2008

Commentationes Mathematicae

Let E be a Banach function space and X be a real Banach space. We study Bochner representable operators from a Köthe-Bochner space E ( X ) to a Banach space Y . We consider the problem of compactness and weak compactness of Bochner representable operators from E ( X ) (provided with the natural mixed topology) to Y .

Linear operators on non-locally convex Orlicz spaces

Marian NowakAgnieszka Oelke — 2008

Banach Center Publications

We study linear operators from a non-locally convex Orlicz space L Φ to a Banach space ( X , | | · | | X ) . Recall that a linear operator T : L Φ X is said to be σ-smooth whenever u ( o ) 0 in L Φ implies | | T ( u ) | | X 0 . It is shown that every σ-smooth operator T : L Φ X factors through the inclusion map j : L Φ L Φ ̅ , where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators T : L Φ X . This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex...

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