On tensor products of nuclear operators in Banach spaces

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

Application of symmetric analytic functions to spectra of linear operators

The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.

The ideal of Lipschitz classical p-compact operators and its injective hull

We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

Involvement of stakeholders during the preparedness phase of post-accident situation management

The Steering Committee for Post-accident Management Preparedness (CODIRPA) was commissioned by the French Government in 2005 with the aim of establishing the main principles to be set up for population protection and recovery in the long term. From the beginning, one of the main principles was the pluralistic nature of the working groups (WGs), including scientific and technical experts, representatives from state departments, nuclear operators, and representatives of civil society (i.e. stakeholders). Stakeholders were mainly associated with the various WGs of CODIRPA. In order to foster the involvement of stakeholders from civil society in the works of CODIRPA, a new organisation was implemented with two WGs: one mainly composed of technical experts for tackling technical issues, and one for evaluating the proposals made by the experts from the stakeholders’ point of view. This article presents the results of this new strategy.

Some results about Lipschitz p-Nuclear Operators

The aim of this paper is to study the onto isometries of the space of strongly Lipschitz p-nuclear operators, introduced by D. Chen and B. Zheng (Nonlinear Anal.,75, 2012). We give some new results about such isometrics and we focus, in particular, on the case F = ℓ p*.

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.