Characterizations of continuous operators on $$C_b(X)$$ with the strict topology

Let X be a completely regular Hausdorff space and $$C_b(X)$$ C b ( X ) be the space of all bounded continuous functions on X, equipped with the strict topology $$\beta $$ β . We study some important classes of $$(\beta ,\Vert \cdot \Vert _E)$$ ( β , ‖ · ‖ E ) -continuous linear operators from $$C_b(X)$$ C b ( X ) to a Banach space $$(E,\Vert \cdot \Vert _E)$$ ( E , ‖ · ‖ E ) : $$\beta $$ β -absolutely summing operators, compact operators and $$\beta $$ β -nuclear operators. We characterize compact operators and $$\beta $$ β -nuclear operators in terms of their representing measures. It is shown that dominated operators and $$\beta $$ β -absolutely summing operators $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E coincide and if, in particular, E has the Radon–Nikodym property, then $$\beta $$ β -absolutely summing operators and $$\beta $$ β -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.

Nuclear operators on Banach function spaces

Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$ ( Ω , Σ , λ ) such that $$L^\infty \subset E\subset L^1$$ L ∞ ⊂ E ⊂ L 1 . Let $$E'$$ E ′ denote the Köthe dual of E and $$\tau (E,E')$$ τ ( E , E ′ ) stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$ T : E → X between the locally convex space $$(E,\tau (E,E'))$$ ( E , τ ( E , E ′ ) ) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$ ( L ∞ , τ ( L ∞ , L 1 ) ) and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$ m T : Σ → X has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$ | m T | ( Ω ) = ‖ T ‖ nuc (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$ L ∞ are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X admits a factorization through some Orlicz space $$L^\varphi $$ L φ , that is, $$T=S\circ i_\infty $$ T = S ∘ i ∞ , where $$S:L^\varphi \rightarrow X$$ S : L φ → X is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$ i ∞ : L ∞ → L φ is the inclusion map.

On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

If \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty, then \omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property.

Uniqueness of norm-preserving extensions of functionals on the space of compact operators

Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^\ast $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^\ast $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^\ast $ is an extreme point of $B_{Y^{\ast} }$ having a unique norm-preserving extension to $Z$, and $x^{\ast\ast} \in B_{X^{\ast\ast} }$, then the only norm-preserving extension of the functional $x^{\ast\ast} \otimes y^\ast \in \mathcal {K}(X,Y)^\ast $ to $\mathcal {K}(X,Z)$ is $x^{\ast\ast} \otimes z^\ast $ where $z^\ast \in Z^\ast $ is the only norm-preserving extension of $y^\ast $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{\ast\ast} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{\ast\ast} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{\ast\ast} $ has the Radon-Nikodým property.

Measures on effect algebras

We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.